Pigenhole Principle Problem

Question

I am doing some hw but I cannot figure out this one. Hint: it is part of a pigeonhole principle.

Question: Prove that if $a$ is a natural number, then there exist two unequal natural numbers $k$ and $l$ for which $a^k - a^l$ is divisible by 10.

Any help much appreciated!

Thank you

Thank you, I corrected the problem, I intended it to be $a^k - a^l$. — Tsangares, Dec 03 '16 at 02:44

AlgorithmsX · Answer 1 · 2016-12-03T02:47:21.827

2

There are only ten possible values for $a^k\bmod10$ for all natural numbers $a$, but infinitely possible values for $k$.

edited Dec 03 '16 at 02:47

answered Dec 03 '16 at 02:45

AlgorithmsX

4,560

Use \bmod for the binary operation; that way you get the right spacing. – Brian M. Scott Dec 03 '16 at 02:47
@BrianM.Scott Thank you for that. I have been using \mod for so long. – AlgorithmsX Dec 03 '16 at 02:47

Pigenhole Principle Problem

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