Competition problem — Number theory

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Number theory Difficulty 6.0 AIME, harder Prove it

21. Let $n, h$ be positive integers. Prove: Among the positive integers not exceeding $n h$ , the number of integers that are coprime to $n$ is equal to $h \varphi(n)$ .

Solution

21. For the given $t, t m+1, t m+2, \cdots, t m+m$ there are exactly $\varphi(m)$ numbers that are coprime with $m$ .

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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.