# What is the significance of $(n,m)=1$ in this proof that the Euler phi function is multiplicative?

In the proof reported in Aaron Grecius's notes on Euler’s Phi Function (p. 1) where is the fact that $$(n,m)=1$$ actually used ? I'm having trouble understanding it's significance.

• All the residue classes $\bmod 6$ are of the form $2a+3b$ with $a \in 1 \ldots 3, b \in 1 \ldots 2$. Not all the residue classes $\bmod 8$ are of the form $2a+4b$ with $a \in 1 \ldots 4, b \in 1 \ldots 2$. To complete the proof, when is $2a+3b$ coprime with $6$ ? – reuns Jul 16 at 9:35
• I don't like the proof presented there. I like to see this theorem as a corollary of the CRT. – user661541 Jul 16 at 12:34

Its wrong otherwise, since e.g., $$\phi(p^n) =(p-1)p^{n-1}$$ and so $$\phi(p^2) = (p-1)p$$, but $$\phi (p)\phi(p) =(p-1)^2$$.