$\phi(n)$ is even for $n>2$

Prove that $\varphi(n)$ is even for every integer $n>2$.

Solution

Write \[ n=\prod_{i=1}^r p_i^{a_i}. \] Then \[ \varphi(n)=\prod_{i=1}^r p_i^{a_i-1}(p_i-1). \]

If some prime factor $p_i$ is odd, then $p_i-1$ is even, so the product is even.

Otherwise the only prime divisor of $n$ is $2$, so $n=2^a$. Since $n>2$, one has $a\ge2$, and \[ \varphi(n)=\varphi(2^a)=2^{a-1}, \] which is even.

Thus $\varphi(n)$ is even for every $n>2$.

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