Euler's totient function of $n^2$

Question

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

Answer 1

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

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

Answer 2

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

Answer 3

Hint: Using this identity

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