Competition problem — Number theory

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Number theory Difficulty 5.4 AIME, harder Find the answer

6. Find all positive integers $n$ that satisfy the following equations:
(i) $\varphi(n)=\varphi(2 n)$ ;
(ii) $\varphi(2 n)=\varphi(3 n)$ ;
(iii) $\varphi(3 n)=\varphi(4 n)$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

6. (i) $2 \nmid n$ ; (ii) $2 \mid n, 3 \nmid n$ ; (iii) $2 \nmid n, 3 \nmid n$ . In (ii), (iii) only discuss
$n=2^{a} 3^{\beta} \cdot m, \quad(6, m)=1$

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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.