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.
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By definition of $\phi$, $\phi(n)=\phi(3^a\cdot 2^b)=2\cdot3^{a-1}\cdot 2^{b-1}=3^{a-1}\cdot 2^b$. thus $\phi(n)|n$/ – Qurultay Feb 23 '20 at 20:48
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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 $\phi (3^a)\cdot \phi (2^b)=3^{a-1}(3-1)2^{b-1}(2-1)=\frac{n}{3}$.
Surajit
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