Why is it that for a,b∈N $φ(3^a2^b) | n$?

Question

I'm having a hard time understanding Euler's totient function. I want to know why when $n=3^a2^b$ for $a,b∈ \mathbb N$ then $\phi(n)|n$? Any help would be greatly appreciated.

Answer 1

If $n=3^a\cdot 2^b$, then $\phi (3^a\cdot 2^b)=\phi (3^a)\cdot \phi (2^b)$ as $\gcd(3^a,2^b)=1$, and $\phi (3^a)\cdot \phi (2^b)=3^{a-1}(3-1)2^{b-1}(2-1)=\frac{n}{3}$.