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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.

Andrés E. Caicedo
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Amr
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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 04:06
  • See Remark 3.3 of http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/eulerthm.pdf, especially equation (3.2). Look also at the two references mentioned at the end of the remark. – KCd Dec 19 '14 at 20:11

1 Answers1

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Are you looking for something like the following?

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

Zz'Rot
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