Modulo Arithmetic: Is a^{\varphi(n)}\equiv 1 (mod \;n) for gcd(a,n)=1?

[Ted123]

Is it true that if A \equiv B \mod{\varphi(N)} where \varphi (N) is Euler's totient function then a^A \equiv a^B \mod{N}?

I'm not after a proof or anything but I didn't do a number theory course and it seems that this fact is used in many questions I'm currently doing.

 

[Hurkyl]

You need to assume gcd(a,N)=1 as well.

 

[johnqwertyful]

The more popular format is

a^{\varphi(n)}\equiv 1 (mod \;n) where gcd(a,n)=1