Hey I would like to show that
$a\mid b\Rightarrow \varphi(a)\mid\varphi(b)\qquad a,b\in\mathbb{N}$
where $\varphi(n)$ is the the totient function.
My try:
Let $a,b\in\mathbb{N}$ and $a\mid b$. By Eulers formula we have that $$ \varphi(a)=a\cdot \prod_{\substack{p \mid a\\ p\ \text{prime}}} \Big(1-\frac{1}{p}\Big). $$ Now since $a\mid b$ by the Fundamental Theorem of Arithmetic we can factorise $a$ and $b$ into $$ a=\prod_{i=1}^{n} p_i^{\alpha_i} \qquad b=\prod_{i=1}^{m} p_i^{\beta_i} $$ with $p_i$ being primes, $n \le m$, and $\alpha_i \le \beta_i$ for $1 \le i \le n$. Using this we can say $$ \left.\prod_{i=1}^{n} \Big(1-\frac{1}{p_i}\Big) \middle| \prod_{i=1}^{n}\Big(1-\frac{1}{p_i}\Big)\right. $$ and this concludes the result.
Now my question is that is this correct? If it is is there another way to arrive to this conclusion? My way of describing this felt a bit clucky.
Happy new years eve to all!
