Question 101
Let $a, b$ be two integers such that $\gcd(a, b)$ has at least two distinct prime factors. Let $S=\{x \in \mathbb{N} \mid x \equiv a \pmod{b}\}$. An element of $S$ is called irreducible if it cannot be written as a product of at least two elements of $S$ (not necessarily distinct).
Show that there exists $N\gt 0$ such that every element of $S$ can be written as a product of at most $N$ irreducible elements of $S$ (not necessarily distinct).
Show that there exists $N\gt 0$ such that every element of $S$ can be written as a product of at most $N$ irreducible elements of $S$ (not necessarily distinct).