Competition problem — Number theory

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Number theory Difficulty 5.7 AIME, harder Prove it

3 Let $m$ be a positive integer greater than 1, and $m \mid (m-1)! + 1$ . Prove: $m$ is a prime number.

Solution

3. If $m$ is a composite number, then there exists a positive integer $p$ , such that $2 \leqslant p<m$ , and $p \mid m$ . In this case, $p \mid (m-1)!$ , but $m \mid (m-1)!+1$ , so $p \mid (m-1)!+1$ , which leads to $p \mid 1$ , a contradiction.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.