Let $a,b,n$ be any integers greater than $1$ and $b >\phi(n),$ where $\phi$ is Euler's totient function; is the below equation is true or not? $$ a^b \equiv_n a^{\phi(n) + (b \mod \phi(n))}$$
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To clarify, you are interested in the general situation where $\gcd(a,n)$ might not equal $1$? – Greg Martin May 15 '19 at 17:32
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If gcd($a,n)=1$ then $a^{\phi(n)}\equiv_n1$ – J. W. Tanner May 15 '19 at 17:42
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@greg-martin, yes I'm interested in the general situation even more than square free generalization. – pistachio May 15 '19 at 17:47
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Duplicate of Proof of : $x^{n}\equiv x^{\varphi(m)+[n \bmod \varphi(m)]} \mod m$ – Bill Dubuque May 15 '19 at 17:52
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@BillDubuque thank you. – pistachio May 15 '19 at 17:58