When does $ ϕ(3n)=3ϕ(n) $?

Question

I think this only works for when $3\mid n$, but I'm trying to find a way to prove this.

I have tried the following:

• Assume $3$ does not divide $n$. Then $\varphi(3n)=3n$. (But how can I show this is not equal to $3\varphi(n)$?)

• If $3\mid n$, then $3n= 3^{\alpha+1}$. (But how can I show this is equal to $3\varphi(n)$?

Answer 1

One direction: Suppose that $3$ does not divide $n$. Then $\varphi(3n)=\varphi(3)\varphi(n)=2\varphi(n)$. Thus $\varphi(3n)\ne 3\varphi(n)$.

The other direction: Suppose that $3$ divides $n$. Let $n=3^km$, where $m$ is not divisible by $3$, and $k\ge 1$.

Then $\varphi(n)=(2)3^{k-1}\varphi(m)$.

Also, $3n=3^{k+1}m$, so $\varphi(3n)=(2)(3^k)\varphi(m)=3\varphi(n)$.

Remark: We used repeatedly the fact that $\varphi$ is multiplicative. If $a$ and $b$ are relatively prime, then $\varphi(ab)=\varphi(a)\varphi(b)$.

We also used the fact that if $p^t$, with $t\ge 1$ is a power of the prime $p$, then $\varphi(p^t)=(p-1)p^{t-1}$. (We only needed it for $p=3$.)