$n, ab \ne 0, a \ne 0 (\mod n), b \ne 0 (\mod n)$

Question

In modular arighmetic, what is the property of a modulus $n$ such that there is no pair $(a, b), a \ne 0, b \ne 0$ where $ab\mod n = 0$. One such modulus is 2. Another is 3.

Answer 1

They are called prime numbers. Short proof:

If $n=pq$, (i.e. $n$ is composite), where $1 < p,q < n$. Then $pq \pmod n = n \pmod n = 0$.