Show that a given term is divisible by a given number

Asked Nov 17 '23 at 14:17

Active Nov 17 '23 at 14:17

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Let $k$ be an odd natural number. Show that the sum $\sum_{i=1}^{500} i^k$ is divisible by $501$.

This means that:

$\sum\limits_{i=1}^{500} i^k \equiv 0 \pmod{501}$

I have researched on the internet for quite some time but don't know how to continue from here. Any help is appreciated!

asked Nov 17 '23 at 14:17

Mike M.

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Hint: what is $i^k + (501 - i)^k \pmod {501}$? – mihaild Nov 17 '23 at 14:20
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Very similar, https://math.stackexchange.com/q/427744/925076 – Dheeraj Gujrathi Nov 17 '23 at 14:28
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Use the reflection symmetry of odd functions as in this $2105$ example in the linked dupe. – Bill Dubuque Nov 17 '23 at 19:40
Thanks everyone, I see where this is going. I just do not understand why k needs to be odd? I understood that in other examples k needed to be odd to avoid making negative i positive. However, in this task, 0 < i < 501, hence i cannot be negative? – Mike M. Nov 18 '23 at 07:28
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@MikeM The pairing up and cancelling of opposite sign terms depends crucially on the fact that $,f(x) = x^k,$ is odd, i.e. $f(-x) = - f(x),,$ which is true iff $k$ is odd. $\ \ $ – Bill Dubuque Nov 18 '23 at 23:16

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