Is $\binom{N}{b}\equiv a\mod N,$ $a\neq 0$, only possibly when $\gcd(N,b)>1$?

Question

Let $0\leq b<N$. If $N$ is prime, then $\binom{N}{b}\equiv 0\mod N$ for all such $b$. For $N$ composite, can it be shown that $\binom{N}{b}\equiv a\mod N,$ $a\neq 0$, is only possible for $\gcd(b,N)>1$?

Answer 1

Hint: this follows from Kummer's theorem on binomial coefficients.