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Find the number of ways to express 1050 as sum of consecutive integers


Proof that the sum of the cubes of any three consecutive positive integers is divisible by three.Number of ways to express $N$ as a sum of consecutive integersCombinatorics question about choosing non consecutive integersNumber of ways to write n as a sum of consecutive integersThe smallest number that can be written as the sum of two or more consecutive integers in exactly 1000 waysHow many ways to write sum as $k$ restricted integersIn how many ways 1387 can written in the sum of $n,(n>2)$ Consecutive natural numbersHow many ways can 15000 be written as the product of 3 integers?Represent Integer as sum of at least two consecutive postive IntegersProve that any number that is not a power of 2 can be expressed as a sum of two or more consecutive positive integers













10












$begingroup$


I have to solve this task:




Find the number of ways to present $1050$ as sum of consecutive
positive integers.




I was thinking if factorization can help there:
$$1050 = 2 cdot 3 cdot 5^2 cdot 7 $$
but I am not sure how to use that information (if there is a sense)



example



I can solve something similar but on smaller scale:



begin{align} 15 &= 15 \ &= 7+8 \ &=4+5+6 \ &= 1+2+3+4+5 end{align}



($4$ ways)










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    so the next step is to work out what you're doing in general. To get $15=7+8$ you are dividing by $2$ and hoping not to get an integer so that you can take the integers either side. If that doesn't work you'd look to divide by $3$ to get three integers, etc. Can you take it from there?
    $endgroup$
    – postmortes
    16 hours ago










  • $begingroup$
    @postmortes chmmm I think that if I divide by odd number ($m$) and I get integers, then I should take $(m-1)/2$ integers from right and the same number integers from left site. But if I divide by even number ($m$) and get non-int then I should get two numbers?
    $endgroup$
    – VirtualUser
    16 hours ago










  • $begingroup$
    Yes, and then you can repeat that with each of the smaller numbers and look for a formula to work out how many times you can do it
    $endgroup$
    – postmortes
    16 hours ago










  • $begingroup$
    but for example $3> 16/6 > 2$ and 16 is not equal to 2+3
    $endgroup$
    – VirtualUser
    16 hours ago










  • $begingroup$
    if you're dividing by $6$ you're looking to represent 16 as the sum of 6 consecutive integers (which you can't do since $1+2+3+4+5=15$).
    $endgroup$
    – postmortes
    16 hours ago
















10












$begingroup$


I have to solve this task:




Find the number of ways to present $1050$ as sum of consecutive
positive integers.




I was thinking if factorization can help there:
$$1050 = 2 cdot 3 cdot 5^2 cdot 7 $$
but I am not sure how to use that information (if there is a sense)



example



I can solve something similar but on smaller scale:



begin{align} 15 &= 15 \ &= 7+8 \ &=4+5+6 \ &= 1+2+3+4+5 end{align}



($4$ ways)










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    so the next step is to work out what you're doing in general. To get $15=7+8$ you are dividing by $2$ and hoping not to get an integer so that you can take the integers either side. If that doesn't work you'd look to divide by $3$ to get three integers, etc. Can you take it from there?
    $endgroup$
    – postmortes
    16 hours ago










  • $begingroup$
    @postmortes chmmm I think that if I divide by odd number ($m$) and I get integers, then I should take $(m-1)/2$ integers from right and the same number integers from left site. But if I divide by even number ($m$) and get non-int then I should get two numbers?
    $endgroup$
    – VirtualUser
    16 hours ago










  • $begingroup$
    Yes, and then you can repeat that with each of the smaller numbers and look for a formula to work out how many times you can do it
    $endgroup$
    – postmortes
    16 hours ago










  • $begingroup$
    but for example $3> 16/6 > 2$ and 16 is not equal to 2+3
    $endgroup$
    – VirtualUser
    16 hours ago










  • $begingroup$
    if you're dividing by $6$ you're looking to represent 16 as the sum of 6 consecutive integers (which you can't do since $1+2+3+4+5=15$).
    $endgroup$
    – postmortes
    16 hours ago














10












10








10


0



$begingroup$


I have to solve this task:




Find the number of ways to present $1050$ as sum of consecutive
positive integers.




I was thinking if factorization can help there:
$$1050 = 2 cdot 3 cdot 5^2 cdot 7 $$
but I am not sure how to use that information (if there is a sense)



example



I can solve something similar but on smaller scale:



begin{align} 15 &= 15 \ &= 7+8 \ &=4+5+6 \ &= 1+2+3+4+5 end{align}



($4$ ways)










share|cite|improve this question











$endgroup$




I have to solve this task:




Find the number of ways to present $1050$ as sum of consecutive
positive integers.




I was thinking if factorization can help there:
$$1050 = 2 cdot 3 cdot 5^2 cdot 7 $$
but I am not sure how to use that information (if there is a sense)



example



I can solve something similar but on smaller scale:



begin{align} 15 &= 15 \ &= 7+8 \ &=4+5+6 \ &= 1+2+3+4+5 end{align}



($4$ ways)







combinatorics elementary-number-theory discrete-mathematics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 12 hours ago









Ennar

14.7k32445




14.7k32445










asked 16 hours ago









VirtualUserVirtualUser

991115




991115








  • 1




    $begingroup$
    so the next step is to work out what you're doing in general. To get $15=7+8$ you are dividing by $2$ and hoping not to get an integer so that you can take the integers either side. If that doesn't work you'd look to divide by $3$ to get three integers, etc. Can you take it from there?
    $endgroup$
    – postmortes
    16 hours ago










  • $begingroup$
    @postmortes chmmm I think that if I divide by odd number ($m$) and I get integers, then I should take $(m-1)/2$ integers from right and the same number integers from left site. But if I divide by even number ($m$) and get non-int then I should get two numbers?
    $endgroup$
    – VirtualUser
    16 hours ago










  • $begingroup$
    Yes, and then you can repeat that with each of the smaller numbers and look for a formula to work out how many times you can do it
    $endgroup$
    – postmortes
    16 hours ago










  • $begingroup$
    but for example $3> 16/6 > 2$ and 16 is not equal to 2+3
    $endgroup$
    – VirtualUser
    16 hours ago










  • $begingroup$
    if you're dividing by $6$ you're looking to represent 16 as the sum of 6 consecutive integers (which you can't do since $1+2+3+4+5=15$).
    $endgroup$
    – postmortes
    16 hours ago














  • 1




    $begingroup$
    so the next step is to work out what you're doing in general. To get $15=7+8$ you are dividing by $2$ and hoping not to get an integer so that you can take the integers either side. If that doesn't work you'd look to divide by $3$ to get three integers, etc. Can you take it from there?
    $endgroup$
    – postmortes
    16 hours ago










  • $begingroup$
    @postmortes chmmm I think that if I divide by odd number ($m$) and I get integers, then I should take $(m-1)/2$ integers from right and the same number integers from left site. But if I divide by even number ($m$) and get non-int then I should get two numbers?
    $endgroup$
    – VirtualUser
    16 hours ago










  • $begingroup$
    Yes, and then you can repeat that with each of the smaller numbers and look for a formula to work out how many times you can do it
    $endgroup$
    – postmortes
    16 hours ago










  • $begingroup$
    but for example $3> 16/6 > 2$ and 16 is not equal to 2+3
    $endgroup$
    – VirtualUser
    16 hours ago










  • $begingroup$
    if you're dividing by $6$ you're looking to represent 16 as the sum of 6 consecutive integers (which you can't do since $1+2+3+4+5=15$).
    $endgroup$
    – postmortes
    16 hours ago








1




1




$begingroup$
so the next step is to work out what you're doing in general. To get $15=7+8$ you are dividing by $2$ and hoping not to get an integer so that you can take the integers either side. If that doesn't work you'd look to divide by $3$ to get three integers, etc. Can you take it from there?
$endgroup$
– postmortes
16 hours ago




$begingroup$
so the next step is to work out what you're doing in general. To get $15=7+8$ you are dividing by $2$ and hoping not to get an integer so that you can take the integers either side. If that doesn't work you'd look to divide by $3$ to get three integers, etc. Can you take it from there?
$endgroup$
– postmortes
16 hours ago












$begingroup$
@postmortes chmmm I think that if I divide by odd number ($m$) and I get integers, then I should take $(m-1)/2$ integers from right and the same number integers from left site. But if I divide by even number ($m$) and get non-int then I should get two numbers?
$endgroup$
– VirtualUser
16 hours ago




$begingroup$
@postmortes chmmm I think that if I divide by odd number ($m$) and I get integers, then I should take $(m-1)/2$ integers from right and the same number integers from left site. But if I divide by even number ($m$) and get non-int then I should get two numbers?
$endgroup$
– VirtualUser
16 hours ago












$begingroup$
Yes, and then you can repeat that with each of the smaller numbers and look for a formula to work out how many times you can do it
$endgroup$
– postmortes
16 hours ago




$begingroup$
Yes, and then you can repeat that with each of the smaller numbers and look for a formula to work out how many times you can do it
$endgroup$
– postmortes
16 hours ago












$begingroup$
but for example $3> 16/6 > 2$ and 16 is not equal to 2+3
$endgroup$
– VirtualUser
16 hours ago




$begingroup$
but for example $3> 16/6 > 2$ and 16 is not equal to 2+3
$endgroup$
– VirtualUser
16 hours ago












$begingroup$
if you're dividing by $6$ you're looking to represent 16 as the sum of 6 consecutive integers (which you can't do since $1+2+3+4+5=15$).
$endgroup$
– postmortes
16 hours ago




$begingroup$
if you're dividing by $6$ you're looking to represent 16 as the sum of 6 consecutive integers (which you can't do since $1+2+3+4+5=15$).
$endgroup$
– postmortes
16 hours ago










6 Answers
6






active

oldest

votes


















13












$begingroup$

We want to find the number of solutions of
$$n+(n+1)+ldots + (n+k) = 1050, ninmathbb Z_{>0}, kinmathbb Z_{geq 0}.tag{1}$$



Rewrite the sum as $$n(k+1) + 0 + 1 +ldots + k = n(k+1) + frac{k(k+1)}{2}= frac 12(2n+k)(k+1).$$



Thus, the number of solutions to $(1)$ is the same as the number of solutions of
$$(2n+k)(k+1) = 2100, ninmathbb Z_{>0}, kinmathbb Z_{geq 0}.tag{2}$$



Let $a$ and $b$ be divisors of $2100$ such that
begin{align}
2n+k &= a,\
k+1 &= b.tag{3}
end{align}

Solving it we get
begin{align}
n &= frac{a-b+1}2,\
k &= b -1.tag{4}
end{align}



From here we see that not every choice of integers $a$ and $b$ such that $ab = 2100$ will give us a solution to $(2)$. Since $a-b+1$ must be even, $a$ and $b$ are of opposite parities. Also, $ageq b > 0$ since $n> 0$ and $k geq 0$.



First determine the number of ways to factor $2100 = 2^2cdot 3cdot 5^2 cdot 7$ such that one of the factors is odd. For this to be fulfilled, we shouldn't allow $4 = 2^2$ to be factored, so consider $2100 = 4cdot 3 cdot 5^2 cdot 7$ instead. Thus, there are $2cdot 2cdot 3cdot 2 = 24$ positive integral solutions to $2100 = a'b'$ such that one factor is odd. Because of commutativity, it means there are $12$ distinct ways to factor $2100$ into product of two factors, one of which is odd, and for every such factorization there is a unique choice for $a$ and $b$ such that $ageq b$.



Thus, there are $12$ positive integral solutions to $(2)$.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    I believe this argument generalizes to show that the number of ways to write any number $n$ as consecutive positive integers is $prod (e_i+1)$, where the $e_i$ are the exponents of all odd primes in $n$'s prime factorization.
    $endgroup$
    – eyeballfrog
    4 hours ago






  • 1




    $begingroup$
    @eyeballfrog, correct.
    $endgroup$
    – Ennar
    4 hours ago



















2












$begingroup$

The sum of the first $n$ natural numbers is
$$sum_{i=0}^ni=frac12n(n+1).$$
So by subtracting the first $m-1$ terms we get the sum of all consecutive integers from $m$ to $n$;
$$sum_{i=m}^ni=frac12n(n+1)-frac12(m-1)m=frac12(n+m)(n-m+1).$$
To count the number of ways to write a number $k$ as a sum of consecutive integers, we want to find natural numbers $m$ and $n$ with $m<n$ such that
$$(n+m)(n-m+1)=2k.$$
In particular this gives a factorization of $2k$. Conversely, if $2k=atimes b$ is a factorization where $anotequiv bpmod{2}$ then setting
$$m:=frac{a-b+1}{2}qquadtext{ and }qquad n:=frac{a+b-1}{2},$$
gives $(n+m)(n-m+1)=2k$. This shows that if $k=2^lk'$ with $linBbb{N}$ and $k'$ odd, then the expressions of $k$ as a sum of consecutive integers correspond $2$-to-$1$ to the divisors of $k'$; for each divisor $d$ of $k'$ we have the two factorizations
$$2k=dtimesleft(2^lfrac{k'}{d}right)=left(2^ldright)timesfrac{k'}{d},$$
of $2k$ into an even and an odd number. The corresponding sums include the trivial sum $k=sum_{i=k}^ki$, as well as sums with negative integers. This shows that the total number of ways to represent a number $k$ as a sum of consecutive integers, is twice the number of divisors of $k'$.



The number of expressions of $k$ as a sum of positive integers is the number of factorizations for which $mgeq0$, or equivalently $a+1geq b$. Of course for every factorization $2k=atimes b$ with $aneq b$ we have either $a+1geq b$ or $b+1geq a$ exclusively, so if $2k$ is not a square then precisely half of all expressions involve only positive integers.



In this particular case $k=1050=2cdot3cdot5^2cdot7$ and so $k'=525=3cdot5^2cdot7$, and the number of divisors of $k'$ equals $2times3times2=12$, so there are $12$ expressions of $k$ as a sum of positive integers. The factors of $k'$ and corresponding sums are
begin{eqnarray*}
text{factor}&&qquad&&text{sums}\
hline
1&&qquad&&k=sum_{i=1050}^{1050}i
&&qquad&&k=sum_{i=261}^{264}i\
3&&qquad&&k=sum_{i=349}^{352}i
&&qquad&&k=sum_{i=82}^{93}i\
5&&qquad&&k=sum_{i=208}^{212}i
&&qquad&&k=sum_{i=43}^{62}i\
7&&qquad&&k=sum_{i=147}^{153}i
&&qquad&&k=sum_{i=24}^{51}i\
15&&qquad&&k=sum_{i=63}^{77}i
&&qquad&&k=sum_{i=-12}^{47}i\
21&&qquad&&k=sum_{i=40}^{60}i
&&qquad&&k=sum_{i=-29}^{54}i\
25&&qquad&&k=sum_{i=30}^{54}i
&&qquad&&k=sum_{i=-39}^{60}i\
35&&qquad&&k=sum_{i=13}^{47}i
&&qquad&&k=sum_{i=-62}^{77}i\
75&&qquad&&k=sum_{i=-23}^{51}i
&&qquad&&k=sum_{i=-146}^{153}i\
105&&qquad&&k=sum_{i=-42}^{62}i
&&qquad&&k=sum_{i=-207}^{212}i\
175&&qquad&&k=sum_{i=-81}^{93}i
&&qquad&&k=sum_{i=-348}^{351}i\
525&&qquad&&k=sum_{i=-260}^{264}i
&&qquad&&k=sum_{i=-1049}^{1050}i\
end{eqnarray*}

We see that indeed $12$ out of these $24$ expressions involve only positive integers.






share|cite|improve this answer











$endgroup$





















    1












    $begingroup$

    For the sum of next integer we may use formula for the sum of arithmetic sequence



    $$sum_{k=1}^na_k=frac n2(a_1+a_n)$$



    So



    begin{aligned}
    1050 &= frac n2(a_1+a_n) \
    2100 &= n(a_1+a_n) \
    2100&= n(a_1+a_n) \
    2100&= n(a_1+a1+(n-1)) \
    2^2cdot3cdot5^2cdot7&= n(2a_1+n-1)
    end{aligned}



    Now:




    1. If $n$ is even, then $(2a_1+n-1)$ is odd, so
      $$n=2^2cdot3^xcdot5^ycdot7^z$$
      where $x in {0,1},; y in {0,1,2}, ;z in {0,1}$,

      so there are $2 times 3 times 2 = 12$ possibilities for $n$.



    2. If $n$ is odd, then similarly
      $$n=3^xcdot5^ycdot7^z$$



      and we obtain other $12$ possibilities for $n$.




    So there are $24$ solutions altogether, a half o them, i. e. $color{red}{12}$, for only positive integers, because for positive integers must be $a_1 ge 1$, and consequently $(2a_1+n-1) >n$, so in the product $n(2a_1+n-1)$ the first multiplier have be smaller than the second one.






    share|cite|improve this answer











    $endgroup$





















      0












      $begingroup$

      Here's how you use that information:



      1050 has divisors:



      $$1,2,3,5,6,7,10,14,15,21,25,30,35,42,50,70,75,105,150,175,210,350,525,1050$$ You can use the prime factorization to check this. You then say 1050 divided by 3 gives 350 so 349+350+351 adds up to 1050. Time to make sums (odd divisors and 4 thrown in, because it lands on a half integer). This gives you:
      $$begin{eqnarray}1050=349+350+351\1050=261+262+263+264\1050=208+209+210+211+212\1050=147+148+149+150+151+152+153\1050=63+64+65+66+67+68+69+70+71+72+73+74+75+76+77\1050=40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60\1050=30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54end{eqnarray}$$



      Okay, I may be missing a few. It gets the point across though.






      share|cite|improve this answer









      $endgroup$





















        0












        $begingroup$

        Using the MiniZinc solver with Gecode, I got the following $11$ solutions:



        13 .. 47
        24 .. 51
        30 .. 54
        40 .. 60
        43 .. 62
        63 .. 77
        82 .. 93
        147 .. 153
        208 .. 212
        261 .. 264
        349 .. 351


        The model:



        var 1..1050: k0;
        var 1..1050: k1;

        constraint
        (1050 == sum([k0 + k | k in 0..k1]));

        solve satisfy;

        output ["n(k0) .. (k0+k1)"];





        share|cite|improve this answer









        $endgroup$





















          0












          $begingroup$

          One way is to note that if there are an odd number, $2n+1$ of terms with the middle term $k$ then the sum of consecutive terms will add to $(2n+1)k$.



          [Because there are $2n+1$ terms and they average $k$]



          And if there are an even number, $2n$ of terms with the middle two terms $k$ and $k+1$ then the sum of consecutive terms will add to $2n(k + frac 12) = n(2k + 1)$.



          [Because there are $2n$ terms and they average $k+frac 12$]



          But if we can't have negative terms we must have $kge n$.



          And so we can have either:



          $1050 = k(2n+1); k> n$ can be a sum of $2n+1$ consecutive terms centered at $k$ (i.e. $(k-n) + (k-n+1) + ..... +(k+n-1)+ (k+n)$ ) or



          $1050 = n(2k+1); nle k$ can be a sum of $2n$ consecutive terms centered at $k$ and $k+1$ (i.e $(k-n+1)+ ..... + (k+n)$.)



          And so



          $1050 = 1050*1 = k(2n+1) implies 1050 = 1050$; one consecutive term centered at $1050$(maybe allowed)



          $1050 = 2*525 = n(2k+1) implies 1050 = 261+262+263+264$; four consectutive terms centered at $262$ and $263$.



          $1050 = 350*3 = k(2n+1) implies 1050= 349 + 350 + 351$; three consecutive terms centered at $350$.



          etc.



          And we can partition $1050 = even*odd$ in... well....



          $1050 = even*odd = (2*3^a5^b7^c)*(3^{1-a}*5^{b-2}*7^{c-1});a=0,1;b=0,1, 2;c=0, 1$ ...



          That would be in $2*3*2 = 12$ ways.



          i.e.



          $1050 = 1050*1=k(2n+1) = 1050$;



          $1050 = 2*525=n(2k+1) = 261+262+263+264$;



          $1050 = 350*3=k(2n+1) = 349 + 350 + 351$;



          $1050 = 210*5 =k(2n+1)= 208+209+210+211+212$;



          $1050 = 6*175=n(2k+1)= 163 + 164+ ..... + 186 + 187$;



          ......



          $1050 = 30*35= k(2n+1) = 13 + 14 + ..... + 46 + 47$;






          share|cite|improve this answer











          $endgroup$













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            6 Answers
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            6 Answers
            6






            active

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            active

            oldest

            votes






            active

            oldest

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            13












            $begingroup$

            We want to find the number of solutions of
            $$n+(n+1)+ldots + (n+k) = 1050, ninmathbb Z_{>0}, kinmathbb Z_{geq 0}.tag{1}$$



            Rewrite the sum as $$n(k+1) + 0 + 1 +ldots + k = n(k+1) + frac{k(k+1)}{2}= frac 12(2n+k)(k+1).$$



            Thus, the number of solutions to $(1)$ is the same as the number of solutions of
            $$(2n+k)(k+1) = 2100, ninmathbb Z_{>0}, kinmathbb Z_{geq 0}.tag{2}$$



            Let $a$ and $b$ be divisors of $2100$ such that
            begin{align}
            2n+k &= a,\
            k+1 &= b.tag{3}
            end{align}

            Solving it we get
            begin{align}
            n &= frac{a-b+1}2,\
            k &= b -1.tag{4}
            end{align}



            From here we see that not every choice of integers $a$ and $b$ such that $ab = 2100$ will give us a solution to $(2)$. Since $a-b+1$ must be even, $a$ and $b$ are of opposite parities. Also, $ageq b > 0$ since $n> 0$ and $k geq 0$.



            First determine the number of ways to factor $2100 = 2^2cdot 3cdot 5^2 cdot 7$ such that one of the factors is odd. For this to be fulfilled, we shouldn't allow $4 = 2^2$ to be factored, so consider $2100 = 4cdot 3 cdot 5^2 cdot 7$ instead. Thus, there are $2cdot 2cdot 3cdot 2 = 24$ positive integral solutions to $2100 = a'b'$ such that one factor is odd. Because of commutativity, it means there are $12$ distinct ways to factor $2100$ into product of two factors, one of which is odd, and for every such factorization there is a unique choice for $a$ and $b$ such that $ageq b$.



            Thus, there are $12$ positive integral solutions to $(2)$.






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              I believe this argument generalizes to show that the number of ways to write any number $n$ as consecutive positive integers is $prod (e_i+1)$, where the $e_i$ are the exponents of all odd primes in $n$'s prime factorization.
              $endgroup$
              – eyeballfrog
              4 hours ago






            • 1




              $begingroup$
              @eyeballfrog, correct.
              $endgroup$
              – Ennar
              4 hours ago
















            13












            $begingroup$

            We want to find the number of solutions of
            $$n+(n+1)+ldots + (n+k) = 1050, ninmathbb Z_{>0}, kinmathbb Z_{geq 0}.tag{1}$$



            Rewrite the sum as $$n(k+1) + 0 + 1 +ldots + k = n(k+1) + frac{k(k+1)}{2}= frac 12(2n+k)(k+1).$$



            Thus, the number of solutions to $(1)$ is the same as the number of solutions of
            $$(2n+k)(k+1) = 2100, ninmathbb Z_{>0}, kinmathbb Z_{geq 0}.tag{2}$$



            Let $a$ and $b$ be divisors of $2100$ such that
            begin{align}
            2n+k &= a,\
            k+1 &= b.tag{3}
            end{align}

            Solving it we get
            begin{align}
            n &= frac{a-b+1}2,\
            k &= b -1.tag{4}
            end{align}



            From here we see that not every choice of integers $a$ and $b$ such that $ab = 2100$ will give us a solution to $(2)$. Since $a-b+1$ must be even, $a$ and $b$ are of opposite parities. Also, $ageq b > 0$ since $n> 0$ and $k geq 0$.



            First determine the number of ways to factor $2100 = 2^2cdot 3cdot 5^2 cdot 7$ such that one of the factors is odd. For this to be fulfilled, we shouldn't allow $4 = 2^2$ to be factored, so consider $2100 = 4cdot 3 cdot 5^2 cdot 7$ instead. Thus, there are $2cdot 2cdot 3cdot 2 = 24$ positive integral solutions to $2100 = a'b'$ such that one factor is odd. Because of commutativity, it means there are $12$ distinct ways to factor $2100$ into product of two factors, one of which is odd, and for every such factorization there is a unique choice for $a$ and $b$ such that $ageq b$.



            Thus, there are $12$ positive integral solutions to $(2)$.






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              I believe this argument generalizes to show that the number of ways to write any number $n$ as consecutive positive integers is $prod (e_i+1)$, where the $e_i$ are the exponents of all odd primes in $n$'s prime factorization.
              $endgroup$
              – eyeballfrog
              4 hours ago






            • 1




              $begingroup$
              @eyeballfrog, correct.
              $endgroup$
              – Ennar
              4 hours ago














            13












            13








            13





            $begingroup$

            We want to find the number of solutions of
            $$n+(n+1)+ldots + (n+k) = 1050, ninmathbb Z_{>0}, kinmathbb Z_{geq 0}.tag{1}$$



            Rewrite the sum as $$n(k+1) + 0 + 1 +ldots + k = n(k+1) + frac{k(k+1)}{2}= frac 12(2n+k)(k+1).$$



            Thus, the number of solutions to $(1)$ is the same as the number of solutions of
            $$(2n+k)(k+1) = 2100, ninmathbb Z_{>0}, kinmathbb Z_{geq 0}.tag{2}$$



            Let $a$ and $b$ be divisors of $2100$ such that
            begin{align}
            2n+k &= a,\
            k+1 &= b.tag{3}
            end{align}

            Solving it we get
            begin{align}
            n &= frac{a-b+1}2,\
            k &= b -1.tag{4}
            end{align}



            From here we see that not every choice of integers $a$ and $b$ such that $ab = 2100$ will give us a solution to $(2)$. Since $a-b+1$ must be even, $a$ and $b$ are of opposite parities. Also, $ageq b > 0$ since $n> 0$ and $k geq 0$.



            First determine the number of ways to factor $2100 = 2^2cdot 3cdot 5^2 cdot 7$ such that one of the factors is odd. For this to be fulfilled, we shouldn't allow $4 = 2^2$ to be factored, so consider $2100 = 4cdot 3 cdot 5^2 cdot 7$ instead. Thus, there are $2cdot 2cdot 3cdot 2 = 24$ positive integral solutions to $2100 = a'b'$ such that one factor is odd. Because of commutativity, it means there are $12$ distinct ways to factor $2100$ into product of two factors, one of which is odd, and for every such factorization there is a unique choice for $a$ and $b$ such that $ageq b$.



            Thus, there are $12$ positive integral solutions to $(2)$.






            share|cite|improve this answer











            $endgroup$



            We want to find the number of solutions of
            $$n+(n+1)+ldots + (n+k) = 1050, ninmathbb Z_{>0}, kinmathbb Z_{geq 0}.tag{1}$$



            Rewrite the sum as $$n(k+1) + 0 + 1 +ldots + k = n(k+1) + frac{k(k+1)}{2}= frac 12(2n+k)(k+1).$$



            Thus, the number of solutions to $(1)$ is the same as the number of solutions of
            $$(2n+k)(k+1) = 2100, ninmathbb Z_{>0}, kinmathbb Z_{geq 0}.tag{2}$$



            Let $a$ and $b$ be divisors of $2100$ such that
            begin{align}
            2n+k &= a,\
            k+1 &= b.tag{3}
            end{align}

            Solving it we get
            begin{align}
            n &= frac{a-b+1}2,\
            k &= b -1.tag{4}
            end{align}



            From here we see that not every choice of integers $a$ and $b$ such that $ab = 2100$ will give us a solution to $(2)$. Since $a-b+1$ must be even, $a$ and $b$ are of opposite parities. Also, $ageq b > 0$ since $n> 0$ and $k geq 0$.



            First determine the number of ways to factor $2100 = 2^2cdot 3cdot 5^2 cdot 7$ such that one of the factors is odd. For this to be fulfilled, we shouldn't allow $4 = 2^2$ to be factored, so consider $2100 = 4cdot 3 cdot 5^2 cdot 7$ instead. Thus, there are $2cdot 2cdot 3cdot 2 = 24$ positive integral solutions to $2100 = a'b'$ such that one factor is odd. Because of commutativity, it means there are $12$ distinct ways to factor $2100$ into product of two factors, one of which is odd, and for every such factorization there is a unique choice for $a$ and $b$ such that $ageq b$.



            Thus, there are $12$ positive integral solutions to $(2)$.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 13 hours ago

























            answered 13 hours ago









            EnnarEnnar

            14.7k32445




            14.7k32445








            • 1




              $begingroup$
              I believe this argument generalizes to show that the number of ways to write any number $n$ as consecutive positive integers is $prod (e_i+1)$, where the $e_i$ are the exponents of all odd primes in $n$'s prime factorization.
              $endgroup$
              – eyeballfrog
              4 hours ago






            • 1




              $begingroup$
              @eyeballfrog, correct.
              $endgroup$
              – Ennar
              4 hours ago














            • 1




              $begingroup$
              I believe this argument generalizes to show that the number of ways to write any number $n$ as consecutive positive integers is $prod (e_i+1)$, where the $e_i$ are the exponents of all odd primes in $n$'s prime factorization.
              $endgroup$
              – eyeballfrog
              4 hours ago






            • 1




              $begingroup$
              @eyeballfrog, correct.
              $endgroup$
              – Ennar
              4 hours ago








            1




            1




            $begingroup$
            I believe this argument generalizes to show that the number of ways to write any number $n$ as consecutive positive integers is $prod (e_i+1)$, where the $e_i$ are the exponents of all odd primes in $n$'s prime factorization.
            $endgroup$
            – eyeballfrog
            4 hours ago




            $begingroup$
            I believe this argument generalizes to show that the number of ways to write any number $n$ as consecutive positive integers is $prod (e_i+1)$, where the $e_i$ are the exponents of all odd primes in $n$'s prime factorization.
            $endgroup$
            – eyeballfrog
            4 hours ago




            1




            1




            $begingroup$
            @eyeballfrog, correct.
            $endgroup$
            – Ennar
            4 hours ago




            $begingroup$
            @eyeballfrog, correct.
            $endgroup$
            – Ennar
            4 hours ago











            2












            $begingroup$

            The sum of the first $n$ natural numbers is
            $$sum_{i=0}^ni=frac12n(n+1).$$
            So by subtracting the first $m-1$ terms we get the sum of all consecutive integers from $m$ to $n$;
            $$sum_{i=m}^ni=frac12n(n+1)-frac12(m-1)m=frac12(n+m)(n-m+1).$$
            To count the number of ways to write a number $k$ as a sum of consecutive integers, we want to find natural numbers $m$ and $n$ with $m<n$ such that
            $$(n+m)(n-m+1)=2k.$$
            In particular this gives a factorization of $2k$. Conversely, if $2k=atimes b$ is a factorization where $anotequiv bpmod{2}$ then setting
            $$m:=frac{a-b+1}{2}qquadtext{ and }qquad n:=frac{a+b-1}{2},$$
            gives $(n+m)(n-m+1)=2k$. This shows that if $k=2^lk'$ with $linBbb{N}$ and $k'$ odd, then the expressions of $k$ as a sum of consecutive integers correspond $2$-to-$1$ to the divisors of $k'$; for each divisor $d$ of $k'$ we have the two factorizations
            $$2k=dtimesleft(2^lfrac{k'}{d}right)=left(2^ldright)timesfrac{k'}{d},$$
            of $2k$ into an even and an odd number. The corresponding sums include the trivial sum $k=sum_{i=k}^ki$, as well as sums with negative integers. This shows that the total number of ways to represent a number $k$ as a sum of consecutive integers, is twice the number of divisors of $k'$.



            The number of expressions of $k$ as a sum of positive integers is the number of factorizations for which $mgeq0$, or equivalently $a+1geq b$. Of course for every factorization $2k=atimes b$ with $aneq b$ we have either $a+1geq b$ or $b+1geq a$ exclusively, so if $2k$ is not a square then precisely half of all expressions involve only positive integers.



            In this particular case $k=1050=2cdot3cdot5^2cdot7$ and so $k'=525=3cdot5^2cdot7$, and the number of divisors of $k'$ equals $2times3times2=12$, so there are $12$ expressions of $k$ as a sum of positive integers. The factors of $k'$ and corresponding sums are
            begin{eqnarray*}
            text{factor}&&qquad&&text{sums}\
            hline
            1&&qquad&&k=sum_{i=1050}^{1050}i
            &&qquad&&k=sum_{i=261}^{264}i\
            3&&qquad&&k=sum_{i=349}^{352}i
            &&qquad&&k=sum_{i=82}^{93}i\
            5&&qquad&&k=sum_{i=208}^{212}i
            &&qquad&&k=sum_{i=43}^{62}i\
            7&&qquad&&k=sum_{i=147}^{153}i
            &&qquad&&k=sum_{i=24}^{51}i\
            15&&qquad&&k=sum_{i=63}^{77}i
            &&qquad&&k=sum_{i=-12}^{47}i\
            21&&qquad&&k=sum_{i=40}^{60}i
            &&qquad&&k=sum_{i=-29}^{54}i\
            25&&qquad&&k=sum_{i=30}^{54}i
            &&qquad&&k=sum_{i=-39}^{60}i\
            35&&qquad&&k=sum_{i=13}^{47}i
            &&qquad&&k=sum_{i=-62}^{77}i\
            75&&qquad&&k=sum_{i=-23}^{51}i
            &&qquad&&k=sum_{i=-146}^{153}i\
            105&&qquad&&k=sum_{i=-42}^{62}i
            &&qquad&&k=sum_{i=-207}^{212}i\
            175&&qquad&&k=sum_{i=-81}^{93}i
            &&qquad&&k=sum_{i=-348}^{351}i\
            525&&qquad&&k=sum_{i=-260}^{264}i
            &&qquad&&k=sum_{i=-1049}^{1050}i\
            end{eqnarray*}

            We see that indeed $12$ out of these $24$ expressions involve only positive integers.






            share|cite|improve this answer











            $endgroup$


















              2












              $begingroup$

              The sum of the first $n$ natural numbers is
              $$sum_{i=0}^ni=frac12n(n+1).$$
              So by subtracting the first $m-1$ terms we get the sum of all consecutive integers from $m$ to $n$;
              $$sum_{i=m}^ni=frac12n(n+1)-frac12(m-1)m=frac12(n+m)(n-m+1).$$
              To count the number of ways to write a number $k$ as a sum of consecutive integers, we want to find natural numbers $m$ and $n$ with $m<n$ such that
              $$(n+m)(n-m+1)=2k.$$
              In particular this gives a factorization of $2k$. Conversely, if $2k=atimes b$ is a factorization where $anotequiv bpmod{2}$ then setting
              $$m:=frac{a-b+1}{2}qquadtext{ and }qquad n:=frac{a+b-1}{2},$$
              gives $(n+m)(n-m+1)=2k$. This shows that if $k=2^lk'$ with $linBbb{N}$ and $k'$ odd, then the expressions of $k$ as a sum of consecutive integers correspond $2$-to-$1$ to the divisors of $k'$; for each divisor $d$ of $k'$ we have the two factorizations
              $$2k=dtimesleft(2^lfrac{k'}{d}right)=left(2^ldright)timesfrac{k'}{d},$$
              of $2k$ into an even and an odd number. The corresponding sums include the trivial sum $k=sum_{i=k}^ki$, as well as sums with negative integers. This shows that the total number of ways to represent a number $k$ as a sum of consecutive integers, is twice the number of divisors of $k'$.



              The number of expressions of $k$ as a sum of positive integers is the number of factorizations for which $mgeq0$, or equivalently $a+1geq b$. Of course for every factorization $2k=atimes b$ with $aneq b$ we have either $a+1geq b$ or $b+1geq a$ exclusively, so if $2k$ is not a square then precisely half of all expressions involve only positive integers.



              In this particular case $k=1050=2cdot3cdot5^2cdot7$ and so $k'=525=3cdot5^2cdot7$, and the number of divisors of $k'$ equals $2times3times2=12$, so there are $12$ expressions of $k$ as a sum of positive integers. The factors of $k'$ and corresponding sums are
              begin{eqnarray*}
              text{factor}&&qquad&&text{sums}\
              hline
              1&&qquad&&k=sum_{i=1050}^{1050}i
              &&qquad&&k=sum_{i=261}^{264}i\
              3&&qquad&&k=sum_{i=349}^{352}i
              &&qquad&&k=sum_{i=82}^{93}i\
              5&&qquad&&k=sum_{i=208}^{212}i
              &&qquad&&k=sum_{i=43}^{62}i\
              7&&qquad&&k=sum_{i=147}^{153}i
              &&qquad&&k=sum_{i=24}^{51}i\
              15&&qquad&&k=sum_{i=63}^{77}i
              &&qquad&&k=sum_{i=-12}^{47}i\
              21&&qquad&&k=sum_{i=40}^{60}i
              &&qquad&&k=sum_{i=-29}^{54}i\
              25&&qquad&&k=sum_{i=30}^{54}i
              &&qquad&&k=sum_{i=-39}^{60}i\
              35&&qquad&&k=sum_{i=13}^{47}i
              &&qquad&&k=sum_{i=-62}^{77}i\
              75&&qquad&&k=sum_{i=-23}^{51}i
              &&qquad&&k=sum_{i=-146}^{153}i\
              105&&qquad&&k=sum_{i=-42}^{62}i
              &&qquad&&k=sum_{i=-207}^{212}i\
              175&&qquad&&k=sum_{i=-81}^{93}i
              &&qquad&&k=sum_{i=-348}^{351}i\
              525&&qquad&&k=sum_{i=-260}^{264}i
              &&qquad&&k=sum_{i=-1049}^{1050}i\
              end{eqnarray*}

              We see that indeed $12$ out of these $24$ expressions involve only positive integers.






              share|cite|improve this answer











              $endgroup$
















                2












                2








                2





                $begingroup$

                The sum of the first $n$ natural numbers is
                $$sum_{i=0}^ni=frac12n(n+1).$$
                So by subtracting the first $m-1$ terms we get the sum of all consecutive integers from $m$ to $n$;
                $$sum_{i=m}^ni=frac12n(n+1)-frac12(m-1)m=frac12(n+m)(n-m+1).$$
                To count the number of ways to write a number $k$ as a sum of consecutive integers, we want to find natural numbers $m$ and $n$ with $m<n$ such that
                $$(n+m)(n-m+1)=2k.$$
                In particular this gives a factorization of $2k$. Conversely, if $2k=atimes b$ is a factorization where $anotequiv bpmod{2}$ then setting
                $$m:=frac{a-b+1}{2}qquadtext{ and }qquad n:=frac{a+b-1}{2},$$
                gives $(n+m)(n-m+1)=2k$. This shows that if $k=2^lk'$ with $linBbb{N}$ and $k'$ odd, then the expressions of $k$ as a sum of consecutive integers correspond $2$-to-$1$ to the divisors of $k'$; for each divisor $d$ of $k'$ we have the two factorizations
                $$2k=dtimesleft(2^lfrac{k'}{d}right)=left(2^ldright)timesfrac{k'}{d},$$
                of $2k$ into an even and an odd number. The corresponding sums include the trivial sum $k=sum_{i=k}^ki$, as well as sums with negative integers. This shows that the total number of ways to represent a number $k$ as a sum of consecutive integers, is twice the number of divisors of $k'$.



                The number of expressions of $k$ as a sum of positive integers is the number of factorizations for which $mgeq0$, or equivalently $a+1geq b$. Of course for every factorization $2k=atimes b$ with $aneq b$ we have either $a+1geq b$ or $b+1geq a$ exclusively, so if $2k$ is not a square then precisely half of all expressions involve only positive integers.



                In this particular case $k=1050=2cdot3cdot5^2cdot7$ and so $k'=525=3cdot5^2cdot7$, and the number of divisors of $k'$ equals $2times3times2=12$, so there are $12$ expressions of $k$ as a sum of positive integers. The factors of $k'$ and corresponding sums are
                begin{eqnarray*}
                text{factor}&&qquad&&text{sums}\
                hline
                1&&qquad&&k=sum_{i=1050}^{1050}i
                &&qquad&&k=sum_{i=261}^{264}i\
                3&&qquad&&k=sum_{i=349}^{352}i
                &&qquad&&k=sum_{i=82}^{93}i\
                5&&qquad&&k=sum_{i=208}^{212}i
                &&qquad&&k=sum_{i=43}^{62}i\
                7&&qquad&&k=sum_{i=147}^{153}i
                &&qquad&&k=sum_{i=24}^{51}i\
                15&&qquad&&k=sum_{i=63}^{77}i
                &&qquad&&k=sum_{i=-12}^{47}i\
                21&&qquad&&k=sum_{i=40}^{60}i
                &&qquad&&k=sum_{i=-29}^{54}i\
                25&&qquad&&k=sum_{i=30}^{54}i
                &&qquad&&k=sum_{i=-39}^{60}i\
                35&&qquad&&k=sum_{i=13}^{47}i
                &&qquad&&k=sum_{i=-62}^{77}i\
                75&&qquad&&k=sum_{i=-23}^{51}i
                &&qquad&&k=sum_{i=-146}^{153}i\
                105&&qquad&&k=sum_{i=-42}^{62}i
                &&qquad&&k=sum_{i=-207}^{212}i\
                175&&qquad&&k=sum_{i=-81}^{93}i
                &&qquad&&k=sum_{i=-348}^{351}i\
                525&&qquad&&k=sum_{i=-260}^{264}i
                &&qquad&&k=sum_{i=-1049}^{1050}i\
                end{eqnarray*}

                We see that indeed $12$ out of these $24$ expressions involve only positive integers.






                share|cite|improve this answer











                $endgroup$



                The sum of the first $n$ natural numbers is
                $$sum_{i=0}^ni=frac12n(n+1).$$
                So by subtracting the first $m-1$ terms we get the sum of all consecutive integers from $m$ to $n$;
                $$sum_{i=m}^ni=frac12n(n+1)-frac12(m-1)m=frac12(n+m)(n-m+1).$$
                To count the number of ways to write a number $k$ as a sum of consecutive integers, we want to find natural numbers $m$ and $n$ with $m<n$ such that
                $$(n+m)(n-m+1)=2k.$$
                In particular this gives a factorization of $2k$. Conversely, if $2k=atimes b$ is a factorization where $anotequiv bpmod{2}$ then setting
                $$m:=frac{a-b+1}{2}qquadtext{ and }qquad n:=frac{a+b-1}{2},$$
                gives $(n+m)(n-m+1)=2k$. This shows that if $k=2^lk'$ with $linBbb{N}$ and $k'$ odd, then the expressions of $k$ as a sum of consecutive integers correspond $2$-to-$1$ to the divisors of $k'$; for each divisor $d$ of $k'$ we have the two factorizations
                $$2k=dtimesleft(2^lfrac{k'}{d}right)=left(2^ldright)timesfrac{k'}{d},$$
                of $2k$ into an even and an odd number. The corresponding sums include the trivial sum $k=sum_{i=k}^ki$, as well as sums with negative integers. This shows that the total number of ways to represent a number $k$ as a sum of consecutive integers, is twice the number of divisors of $k'$.



                The number of expressions of $k$ as a sum of positive integers is the number of factorizations for which $mgeq0$, or equivalently $a+1geq b$. Of course for every factorization $2k=atimes b$ with $aneq b$ we have either $a+1geq b$ or $b+1geq a$ exclusively, so if $2k$ is not a square then precisely half of all expressions involve only positive integers.



                In this particular case $k=1050=2cdot3cdot5^2cdot7$ and so $k'=525=3cdot5^2cdot7$, and the number of divisors of $k'$ equals $2times3times2=12$, so there are $12$ expressions of $k$ as a sum of positive integers. The factors of $k'$ and corresponding sums are
                begin{eqnarray*}
                text{factor}&&qquad&&text{sums}\
                hline
                1&&qquad&&k=sum_{i=1050}^{1050}i
                &&qquad&&k=sum_{i=261}^{264}i\
                3&&qquad&&k=sum_{i=349}^{352}i
                &&qquad&&k=sum_{i=82}^{93}i\
                5&&qquad&&k=sum_{i=208}^{212}i
                &&qquad&&k=sum_{i=43}^{62}i\
                7&&qquad&&k=sum_{i=147}^{153}i
                &&qquad&&k=sum_{i=24}^{51}i\
                15&&qquad&&k=sum_{i=63}^{77}i
                &&qquad&&k=sum_{i=-12}^{47}i\
                21&&qquad&&k=sum_{i=40}^{60}i
                &&qquad&&k=sum_{i=-29}^{54}i\
                25&&qquad&&k=sum_{i=30}^{54}i
                &&qquad&&k=sum_{i=-39}^{60}i\
                35&&qquad&&k=sum_{i=13}^{47}i
                &&qquad&&k=sum_{i=-62}^{77}i\
                75&&qquad&&k=sum_{i=-23}^{51}i
                &&qquad&&k=sum_{i=-146}^{153}i\
                105&&qquad&&k=sum_{i=-42}^{62}i
                &&qquad&&k=sum_{i=-207}^{212}i\
                175&&qquad&&k=sum_{i=-81}^{93}i
                &&qquad&&k=sum_{i=-348}^{351}i\
                525&&qquad&&k=sum_{i=-260}^{264}i
                &&qquad&&k=sum_{i=-1049}^{1050}i\
                end{eqnarray*}

                We see that indeed $12$ out of these $24$ expressions involve only positive integers.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 13 hours ago

























                answered 13 hours ago









                ServaesServaes

                26.6k34098




                26.6k34098























                    1












                    $begingroup$

                    For the sum of next integer we may use formula for the sum of arithmetic sequence



                    $$sum_{k=1}^na_k=frac n2(a_1+a_n)$$



                    So



                    begin{aligned}
                    1050 &= frac n2(a_1+a_n) \
                    2100 &= n(a_1+a_n) \
                    2100&= n(a_1+a_n) \
                    2100&= n(a_1+a1+(n-1)) \
                    2^2cdot3cdot5^2cdot7&= n(2a_1+n-1)
                    end{aligned}



                    Now:




                    1. If $n$ is even, then $(2a_1+n-1)$ is odd, so
                      $$n=2^2cdot3^xcdot5^ycdot7^z$$
                      where $x in {0,1},; y in {0,1,2}, ;z in {0,1}$,

                      so there are $2 times 3 times 2 = 12$ possibilities for $n$.



                    2. If $n$ is odd, then similarly
                      $$n=3^xcdot5^ycdot7^z$$



                      and we obtain other $12$ possibilities for $n$.




                    So there are $24$ solutions altogether, a half o them, i. e. $color{red}{12}$, for only positive integers, because for positive integers must be $a_1 ge 1$, and consequently $(2a_1+n-1) >n$, so in the product $n(2a_1+n-1)$ the first multiplier have be smaller than the second one.






                    share|cite|improve this answer











                    $endgroup$


















                      1












                      $begingroup$

                      For the sum of next integer we may use formula for the sum of arithmetic sequence



                      $$sum_{k=1}^na_k=frac n2(a_1+a_n)$$



                      So



                      begin{aligned}
                      1050 &= frac n2(a_1+a_n) \
                      2100 &= n(a_1+a_n) \
                      2100&= n(a_1+a_n) \
                      2100&= n(a_1+a1+(n-1)) \
                      2^2cdot3cdot5^2cdot7&= n(2a_1+n-1)
                      end{aligned}



                      Now:




                      1. If $n$ is even, then $(2a_1+n-1)$ is odd, so
                        $$n=2^2cdot3^xcdot5^ycdot7^z$$
                        where $x in {0,1},; y in {0,1,2}, ;z in {0,1}$,

                        so there are $2 times 3 times 2 = 12$ possibilities for $n$.



                      2. If $n$ is odd, then similarly
                        $$n=3^xcdot5^ycdot7^z$$



                        and we obtain other $12$ possibilities for $n$.




                      So there are $24$ solutions altogether, a half o them, i. e. $color{red}{12}$, for only positive integers, because for positive integers must be $a_1 ge 1$, and consequently $(2a_1+n-1) >n$, so in the product $n(2a_1+n-1)$ the first multiplier have be smaller than the second one.






                      share|cite|improve this answer











                      $endgroup$
















                        1












                        1








                        1





                        $begingroup$

                        For the sum of next integer we may use formula for the sum of arithmetic sequence



                        $$sum_{k=1}^na_k=frac n2(a_1+a_n)$$



                        So



                        begin{aligned}
                        1050 &= frac n2(a_1+a_n) \
                        2100 &= n(a_1+a_n) \
                        2100&= n(a_1+a_n) \
                        2100&= n(a_1+a1+(n-1)) \
                        2^2cdot3cdot5^2cdot7&= n(2a_1+n-1)
                        end{aligned}



                        Now:




                        1. If $n$ is even, then $(2a_1+n-1)$ is odd, so
                          $$n=2^2cdot3^xcdot5^ycdot7^z$$
                          where $x in {0,1},; y in {0,1,2}, ;z in {0,1}$,

                          so there are $2 times 3 times 2 = 12$ possibilities for $n$.



                        2. If $n$ is odd, then similarly
                          $$n=3^xcdot5^ycdot7^z$$



                          and we obtain other $12$ possibilities for $n$.




                        So there are $24$ solutions altogether, a half o them, i. e. $color{red}{12}$, for only positive integers, because for positive integers must be $a_1 ge 1$, and consequently $(2a_1+n-1) >n$, so in the product $n(2a_1+n-1)$ the first multiplier have be smaller than the second one.






                        share|cite|improve this answer











                        $endgroup$



                        For the sum of next integer we may use formula for the sum of arithmetic sequence



                        $$sum_{k=1}^na_k=frac n2(a_1+a_n)$$



                        So



                        begin{aligned}
                        1050 &= frac n2(a_1+a_n) \
                        2100 &= n(a_1+a_n) \
                        2100&= n(a_1+a_n) \
                        2100&= n(a_1+a1+(n-1)) \
                        2^2cdot3cdot5^2cdot7&= n(2a_1+n-1)
                        end{aligned}



                        Now:




                        1. If $n$ is even, then $(2a_1+n-1)$ is odd, so
                          $$n=2^2cdot3^xcdot5^ycdot7^z$$
                          where $x in {0,1},; y in {0,1,2}, ;z in {0,1}$,

                          so there are $2 times 3 times 2 = 12$ possibilities for $n$.



                        2. If $n$ is odd, then similarly
                          $$n=3^xcdot5^ycdot7^z$$



                          and we obtain other $12$ possibilities for $n$.




                        So there are $24$ solutions altogether, a half o them, i. e. $color{red}{12}$, for only positive integers, because for positive integers must be $a_1 ge 1$, and consequently $(2a_1+n-1) >n$, so in the product $n(2a_1+n-1)$ the first multiplier have be smaller than the second one.







                        share|cite|improve this answer














                        share|cite|improve this answer



                        share|cite|improve this answer








                        edited 12 hours ago

























                        answered 14 hours ago









                        MarianDMarianD

                        6831611




                        6831611























                            0












                            $begingroup$

                            Here's how you use that information:



                            1050 has divisors:



                            $$1,2,3,5,6,7,10,14,15,21,25,30,35,42,50,70,75,105,150,175,210,350,525,1050$$ You can use the prime factorization to check this. You then say 1050 divided by 3 gives 350 so 349+350+351 adds up to 1050. Time to make sums (odd divisors and 4 thrown in, because it lands on a half integer). This gives you:
                            $$begin{eqnarray}1050=349+350+351\1050=261+262+263+264\1050=208+209+210+211+212\1050=147+148+149+150+151+152+153\1050=63+64+65+66+67+68+69+70+71+72+73+74+75+76+77\1050=40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60\1050=30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54end{eqnarray}$$



                            Okay, I may be missing a few. It gets the point across though.






                            share|cite|improve this answer









                            $endgroup$


















                              0












                              $begingroup$

                              Here's how you use that information:



                              1050 has divisors:



                              $$1,2,3,5,6,7,10,14,15,21,25,30,35,42,50,70,75,105,150,175,210,350,525,1050$$ You can use the prime factorization to check this. You then say 1050 divided by 3 gives 350 so 349+350+351 adds up to 1050. Time to make sums (odd divisors and 4 thrown in, because it lands on a half integer). This gives you:
                              $$begin{eqnarray}1050=349+350+351\1050=261+262+263+264\1050=208+209+210+211+212\1050=147+148+149+150+151+152+153\1050=63+64+65+66+67+68+69+70+71+72+73+74+75+76+77\1050=40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60\1050=30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54end{eqnarray}$$



                              Okay, I may be missing a few. It gets the point across though.






                              share|cite|improve this answer









                              $endgroup$
















                                0












                                0








                                0





                                $begingroup$

                                Here's how you use that information:



                                1050 has divisors:



                                $$1,2,3,5,6,7,10,14,15,21,25,30,35,42,50,70,75,105,150,175,210,350,525,1050$$ You can use the prime factorization to check this. You then say 1050 divided by 3 gives 350 so 349+350+351 adds up to 1050. Time to make sums (odd divisors and 4 thrown in, because it lands on a half integer). This gives you:
                                $$begin{eqnarray}1050=349+350+351\1050=261+262+263+264\1050=208+209+210+211+212\1050=147+148+149+150+151+152+153\1050=63+64+65+66+67+68+69+70+71+72+73+74+75+76+77\1050=40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60\1050=30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54end{eqnarray}$$



                                Okay, I may be missing a few. It gets the point across though.






                                share|cite|improve this answer









                                $endgroup$



                                Here's how you use that information:



                                1050 has divisors:



                                $$1,2,3,5,6,7,10,14,15,21,25,30,35,42,50,70,75,105,150,175,210,350,525,1050$$ You can use the prime factorization to check this. You then say 1050 divided by 3 gives 350 so 349+350+351 adds up to 1050. Time to make sums (odd divisors and 4 thrown in, because it lands on a half integer). This gives you:
                                $$begin{eqnarray}1050=349+350+351\1050=261+262+263+264\1050=208+209+210+211+212\1050=147+148+149+150+151+152+153\1050=63+64+65+66+67+68+69+70+71+72+73+74+75+76+77\1050=40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60\1050=30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54end{eqnarray}$$



                                Okay, I may be missing a few. It gets the point across though.







                                share|cite|improve this answer












                                share|cite|improve this answer



                                share|cite|improve this answer










                                answered 15 hours ago









                                Roddy MacPheeRoddy MacPhee

                                17414




                                17414























                                    0












                                    $begingroup$

                                    Using the MiniZinc solver with Gecode, I got the following $11$ solutions:



                                    13 .. 47
                                    24 .. 51
                                    30 .. 54
                                    40 .. 60
                                    43 .. 62
                                    63 .. 77
                                    82 .. 93
                                    147 .. 153
                                    208 .. 212
                                    261 .. 264
                                    349 .. 351


                                    The model:



                                    var 1..1050: k0;
                                    var 1..1050: k1;

                                    constraint
                                    (1050 == sum([k0 + k | k in 0..k1]));

                                    solve satisfy;

                                    output ["n(k0) .. (k0+k1)"];





                                    share|cite|improve this answer









                                    $endgroup$


















                                      0












                                      $begingroup$

                                      Using the MiniZinc solver with Gecode, I got the following $11$ solutions:



                                      13 .. 47
                                      24 .. 51
                                      30 .. 54
                                      40 .. 60
                                      43 .. 62
                                      63 .. 77
                                      82 .. 93
                                      147 .. 153
                                      208 .. 212
                                      261 .. 264
                                      349 .. 351


                                      The model:



                                      var 1..1050: k0;
                                      var 1..1050: k1;

                                      constraint
                                      (1050 == sum([k0 + k | k in 0..k1]));

                                      solve satisfy;

                                      output ["n(k0) .. (k0+k1)"];





                                      share|cite|improve this answer









                                      $endgroup$
















                                        0












                                        0








                                        0





                                        $begingroup$

                                        Using the MiniZinc solver with Gecode, I got the following $11$ solutions:



                                        13 .. 47
                                        24 .. 51
                                        30 .. 54
                                        40 .. 60
                                        43 .. 62
                                        63 .. 77
                                        82 .. 93
                                        147 .. 153
                                        208 .. 212
                                        261 .. 264
                                        349 .. 351


                                        The model:



                                        var 1..1050: k0;
                                        var 1..1050: k1;

                                        constraint
                                        (1050 == sum([k0 + k | k in 0..k1]));

                                        solve satisfy;

                                        output ["n(k0) .. (k0+k1)"];





                                        share|cite|improve this answer









                                        $endgroup$



                                        Using the MiniZinc solver with Gecode, I got the following $11$ solutions:



                                        13 .. 47
                                        24 .. 51
                                        30 .. 54
                                        40 .. 60
                                        43 .. 62
                                        63 .. 77
                                        82 .. 93
                                        147 .. 153
                                        208 .. 212
                                        261 .. 264
                                        349 .. 351


                                        The model:



                                        var 1..1050: k0;
                                        var 1..1050: k1;

                                        constraint
                                        (1050 == sum([k0 + k | k in 0..k1]));

                                        solve satisfy;

                                        output ["n(k0) .. (k0+k1)"];






                                        share|cite|improve this answer












                                        share|cite|improve this answer



                                        share|cite|improve this answer










                                        answered 9 hours ago









                                        Axel KemperAxel Kemper

                                        3,34611418




                                        3,34611418























                                            0












                                            $begingroup$

                                            One way is to note that if there are an odd number, $2n+1$ of terms with the middle term $k$ then the sum of consecutive terms will add to $(2n+1)k$.



                                            [Because there are $2n+1$ terms and they average $k$]



                                            And if there are an even number, $2n$ of terms with the middle two terms $k$ and $k+1$ then the sum of consecutive terms will add to $2n(k + frac 12) = n(2k + 1)$.



                                            [Because there are $2n$ terms and they average $k+frac 12$]



                                            But if we can't have negative terms we must have $kge n$.



                                            And so we can have either:



                                            $1050 = k(2n+1); k> n$ can be a sum of $2n+1$ consecutive terms centered at $k$ (i.e. $(k-n) + (k-n+1) + ..... +(k+n-1)+ (k+n)$ ) or



                                            $1050 = n(2k+1); nle k$ can be a sum of $2n$ consecutive terms centered at $k$ and $k+1$ (i.e $(k-n+1)+ ..... + (k+n)$.)



                                            And so



                                            $1050 = 1050*1 = k(2n+1) implies 1050 = 1050$; one consecutive term centered at $1050$(maybe allowed)



                                            $1050 = 2*525 = n(2k+1) implies 1050 = 261+262+263+264$; four consectutive terms centered at $262$ and $263$.



                                            $1050 = 350*3 = k(2n+1) implies 1050= 349 + 350 + 351$; three consecutive terms centered at $350$.



                                            etc.



                                            And we can partition $1050 = even*odd$ in... well....



                                            $1050 = even*odd = (2*3^a5^b7^c)*(3^{1-a}*5^{b-2}*7^{c-1});a=0,1;b=0,1, 2;c=0, 1$ ...



                                            That would be in $2*3*2 = 12$ ways.



                                            i.e.



                                            $1050 = 1050*1=k(2n+1) = 1050$;



                                            $1050 = 2*525=n(2k+1) = 261+262+263+264$;



                                            $1050 = 350*3=k(2n+1) = 349 + 350 + 351$;



                                            $1050 = 210*5 =k(2n+1)= 208+209+210+211+212$;



                                            $1050 = 6*175=n(2k+1)= 163 + 164+ ..... + 186 + 187$;



                                            ......



                                            $1050 = 30*35= k(2n+1) = 13 + 14 + ..... + 46 + 47$;






                                            share|cite|improve this answer











                                            $endgroup$


















                                              0












                                              $begingroup$

                                              One way is to note that if there are an odd number, $2n+1$ of terms with the middle term $k$ then the sum of consecutive terms will add to $(2n+1)k$.



                                              [Because there are $2n+1$ terms and they average $k$]



                                              And if there are an even number, $2n$ of terms with the middle two terms $k$ and $k+1$ then the sum of consecutive terms will add to $2n(k + frac 12) = n(2k + 1)$.



                                              [Because there are $2n$ terms and they average $k+frac 12$]



                                              But if we can't have negative terms we must have $kge n$.



                                              And so we can have either:



                                              $1050 = k(2n+1); k> n$ can be a sum of $2n+1$ consecutive terms centered at $k$ (i.e. $(k-n) + (k-n+1) + ..... +(k+n-1)+ (k+n)$ ) or



                                              $1050 = n(2k+1); nle k$ can be a sum of $2n$ consecutive terms centered at $k$ and $k+1$ (i.e $(k-n+1)+ ..... + (k+n)$.)



                                              And so



                                              $1050 = 1050*1 = k(2n+1) implies 1050 = 1050$; one consecutive term centered at $1050$(maybe allowed)



                                              $1050 = 2*525 = n(2k+1) implies 1050 = 261+262+263+264$; four consectutive terms centered at $262$ and $263$.



                                              $1050 = 350*3 = k(2n+1) implies 1050= 349 + 350 + 351$; three consecutive terms centered at $350$.



                                              etc.



                                              And we can partition $1050 = even*odd$ in... well....



                                              $1050 = even*odd = (2*3^a5^b7^c)*(3^{1-a}*5^{b-2}*7^{c-1});a=0,1;b=0,1, 2;c=0, 1$ ...



                                              That would be in $2*3*2 = 12$ ways.



                                              i.e.



                                              $1050 = 1050*1=k(2n+1) = 1050$;



                                              $1050 = 2*525=n(2k+1) = 261+262+263+264$;



                                              $1050 = 350*3=k(2n+1) = 349 + 350 + 351$;



                                              $1050 = 210*5 =k(2n+1)= 208+209+210+211+212$;



                                              $1050 = 6*175=n(2k+1)= 163 + 164+ ..... + 186 + 187$;



                                              ......



                                              $1050 = 30*35= k(2n+1) = 13 + 14 + ..... + 46 + 47$;






                                              share|cite|improve this answer











                                              $endgroup$
















                                                0












                                                0








                                                0





                                                $begingroup$

                                                One way is to note that if there are an odd number, $2n+1$ of terms with the middle term $k$ then the sum of consecutive terms will add to $(2n+1)k$.



                                                [Because there are $2n+1$ terms and they average $k$]



                                                And if there are an even number, $2n$ of terms with the middle two terms $k$ and $k+1$ then the sum of consecutive terms will add to $2n(k + frac 12) = n(2k + 1)$.



                                                [Because there are $2n$ terms and they average $k+frac 12$]



                                                But if we can't have negative terms we must have $kge n$.



                                                And so we can have either:



                                                $1050 = k(2n+1); k> n$ can be a sum of $2n+1$ consecutive terms centered at $k$ (i.e. $(k-n) + (k-n+1) + ..... +(k+n-1)+ (k+n)$ ) or



                                                $1050 = n(2k+1); nle k$ can be a sum of $2n$ consecutive terms centered at $k$ and $k+1$ (i.e $(k-n+1)+ ..... + (k+n)$.)



                                                And so



                                                $1050 = 1050*1 = k(2n+1) implies 1050 = 1050$; one consecutive term centered at $1050$(maybe allowed)



                                                $1050 = 2*525 = n(2k+1) implies 1050 = 261+262+263+264$; four consectutive terms centered at $262$ and $263$.



                                                $1050 = 350*3 = k(2n+1) implies 1050= 349 + 350 + 351$; three consecutive terms centered at $350$.



                                                etc.



                                                And we can partition $1050 = even*odd$ in... well....



                                                $1050 = even*odd = (2*3^a5^b7^c)*(3^{1-a}*5^{b-2}*7^{c-1});a=0,1;b=0,1, 2;c=0, 1$ ...



                                                That would be in $2*3*2 = 12$ ways.



                                                i.e.



                                                $1050 = 1050*1=k(2n+1) = 1050$;



                                                $1050 = 2*525=n(2k+1) = 261+262+263+264$;



                                                $1050 = 350*3=k(2n+1) = 349 + 350 + 351$;



                                                $1050 = 210*5 =k(2n+1)= 208+209+210+211+212$;



                                                $1050 = 6*175=n(2k+1)= 163 + 164+ ..... + 186 + 187$;



                                                ......



                                                $1050 = 30*35= k(2n+1) = 13 + 14 + ..... + 46 + 47$;






                                                share|cite|improve this answer











                                                $endgroup$



                                                One way is to note that if there are an odd number, $2n+1$ of terms with the middle term $k$ then the sum of consecutive terms will add to $(2n+1)k$.



                                                [Because there are $2n+1$ terms and they average $k$]



                                                And if there are an even number, $2n$ of terms with the middle two terms $k$ and $k+1$ then the sum of consecutive terms will add to $2n(k + frac 12) = n(2k + 1)$.



                                                [Because there are $2n$ terms and they average $k+frac 12$]



                                                But if we can't have negative terms we must have $kge n$.



                                                And so we can have either:



                                                $1050 = k(2n+1); k> n$ can be a sum of $2n+1$ consecutive terms centered at $k$ (i.e. $(k-n) + (k-n+1) + ..... +(k+n-1)+ (k+n)$ ) or



                                                $1050 = n(2k+1); nle k$ can be a sum of $2n$ consecutive terms centered at $k$ and $k+1$ (i.e $(k-n+1)+ ..... + (k+n)$.)



                                                And so



                                                $1050 = 1050*1 = k(2n+1) implies 1050 = 1050$; one consecutive term centered at $1050$(maybe allowed)



                                                $1050 = 2*525 = n(2k+1) implies 1050 = 261+262+263+264$; four consectutive terms centered at $262$ and $263$.



                                                $1050 = 350*3 = k(2n+1) implies 1050= 349 + 350 + 351$; three consecutive terms centered at $350$.



                                                etc.



                                                And we can partition $1050 = even*odd$ in... well....



                                                $1050 = even*odd = (2*3^a5^b7^c)*(3^{1-a}*5^{b-2}*7^{c-1});a=0,1;b=0,1, 2;c=0, 1$ ...



                                                That would be in $2*3*2 = 12$ ways.



                                                i.e.



                                                $1050 = 1050*1=k(2n+1) = 1050$;



                                                $1050 = 2*525=n(2k+1) = 261+262+263+264$;



                                                $1050 = 350*3=k(2n+1) = 349 + 350 + 351$;



                                                $1050 = 210*5 =k(2n+1)= 208+209+210+211+212$;



                                                $1050 = 6*175=n(2k+1)= 163 + 164+ ..... + 186 + 187$;



                                                ......



                                                $1050 = 30*35= k(2n+1) = 13 + 14 + ..... + 46 + 47$;







                                                share|cite|improve this answer














                                                share|cite|improve this answer



                                                share|cite|improve this answer








                                                edited 6 hours ago

























                                                answered 6 hours ago









                                                fleabloodfleablood

                                                71.7k22686




                                                71.7k22686






























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