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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.

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  • $\begingroup$ 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 $6$ ? $\endgroup$ – reuns Jul 16 at 9:35
  • $\begingroup$ I don't like the proof presented there. I like to see this theorem as a corollary of the CRT. $\endgroup$ – user661541 Jul 16 at 12:34
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Its wrong otherwise, since e.g., $\phi(p^n) =(p-1)p^{n-1}$ and so $\phi(p^2) = (p-1)p$, but $\phi (p)\phi(p) =(p-1)^2$.

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