Competition problem — Number theory

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Number theory Difficulty 5.0 AIME, harder Prove it Austria

Let $n$ be a positive integer.
Prove that $a(n) = n^5 + 5^n$ is divisible by $11$ if and only if $b(n) = n^5 \cdot 5^n + 1$ is divisible by $11$ .

Solution

If $n$ is a multiple of $11$ , both sides of the equivalence are wrong, so the equivalence is true.

If $n$ is not a multiple of $11$ , Fermat's little theorem implies that $n^{10} - 1$ is a multiple of $11$ . The equivalence now follows from
$n^5 a(n) = n^{10} + n^5 \cdot 5^n \equiv b(n) \pmod{11}.$

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