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Prove that, for any positive integer $n$, $$1^n+2^n+3^n+4^n$$ is divisible by $5

Source: Eotvos 1901 p1

March 19, 2020
algebra

Problem Statement

Prove that, for any positive integer nn, 1n+2n+3n+4n1^n+2^n+3^n+4^n is divisible by 55 if and only if nn is not divisible by 44.
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