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Math Help - sum of squares divisible by 4

  1. #1
    Senior Member nikhil's Avatar
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    Exclamation sum of squares divisible by 4

    if Sm denotes sum of squares of first m natural numbers then for how many m<100 is Sm multiple of 4?
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  2. #2
    MHF Contributor
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    Re: sum of squares divisible by 4

    Quote Originally Posted by nikhil View Post
    if Sm denotes sum of squares of first m natural numbers then for how many m<100 is Sm multiple of 4?
    24

    $\left(
    \begin{array}{cc}
    7 & 140 \\
    8 & 204 \\
    15 & 1240 \\
    16 & 1496 \\
    23 & 4324 \\
    24 & 4900 \\
    31 & 10416 \\
    32 & 11440 \\
    39 & 20540 \\
    40 & 22140 \\
    47 & 35720 \\
    48 & 38024 \\
    55 & 56980 \\
    56 & 60116 \\
    63 & 85344 \\
    64 & 89440 \\
    71 & 121836 \\
    72 & 127020 \\
    79 & 167480 \\
    80 & 173880 \\
    87 & 223300 \\
    88 & 231044 \\
    95 & 290320 \\
    96 & 299536 \\
    \end{array}
    \right)$

    there may be some clever way of doing this that one of our number theory experts can comment on.
    Last edited by romsek; March 2nd 2014 at 05:24 PM.
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  3. #3
    MHF Contributor
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    Re: sum of squares divisible by 4

    there is another way of doing this

    $\displaystyle \sum_{k=1}^n k^2 = \dfrac{n(n+1)(2n+1)}{6}$

    so by solving

    $n(n+1)(2n+1) \mod 24 = 0\mbox{ for }1 < n < 100$ you find your answer

    sum of squares divisible by 4-clipboard01.jpg

    you can note it's the same n's as found by brute force checking.
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