Prime Numbers and Primitive Roots

Prime Numbers and Primitive Roots

Let $p_1$, $ p_2$, $p_3$ different prime numbers. Let $N = p_1p_2p_3$.
Given $(p_1-1)|(N-1), (p_2-1)|(N-1)$ and $(p_3-1)|(N-1)$, prove that for
every number $a \in \Bbb N$ such that $\gcd(a,N) = 1$ , $a^N \equiv a \mod
N $. I'd be grateful if anyone could point me to the solution, thanks in
advance.