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It is well known that if $\gcd(a,n)=1$, then $a^{ϕ(n)}=1$ mod $n$.

Are there any results similar to Euler's theorem that can be used when $a$ and $n$ are not coprime.

Feel free to add any restrictions on $a$, $n$ other than $a$, $n$ being coprime.

Thank you.

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What do you mean by "similar" and "can be used"? If your ultimate motive is to calculate $a^m \bmod n$, then you can easily get $a^m \bmod p^k$ for all the prime powers in the prime factorization of $n$, and use the Chinese remainder theorem to put them together. –  ShreevatsaR Nov 12 '12 at 4:06

1 Answer 1

Are you looking for something like the following?

$$\text{For any integers $a$ and $n$, $a^{n} \equiv a^{n-\varphi(n)} \pmod{n}$.}$$

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