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Prove that $\varphi(n^2)=n\cdot\varphi(n)$ for $n\in \Bbb{N}$, where $\varphi$ is Euler's totient function.

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closed as off-topic by kingW3, Henrik, mrp, Davide Giraudo, Namaste Jun 17 '17 at 21:12

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  • $\begingroup$ Formula $\mapsto$ LHS = RHS $\endgroup$ – user228113 Jun 17 '17 at 19:25
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Hint: Use that for any $\;n=p_1^{a_1}\cdot\ldots\cdot p_k^{a_k}\in\Bbb N\;,\;\;p_i\;$ primes, $\;a_i\in\Bbb N\;$ , we have

$$\varphi(n)=n\prod_{i=1}^k\left(1-\frac1{p_i}\right)$$

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$Hint:$ Use multiplicity of Euler's function and fundamental theorem of arithmetics.

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Hint: Using this identity

$$\varphi(n^m)=n^{m-1}\varphi(n)$$

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