Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Prove it

Example 4. If all terms of an arithmetic sequence are integers, and there are $p$ terms, if each term is coprime with $p$ , then its common difference must not be coprime with $p$ .

Solution

〔Hint〕Consider all terms
$a, a+d, a+2d, \cdots, a+(p-1)d$

There must be two terms congruent modulo $p$ . Consider the difference between these terms.

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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.