Question

Without using Theorem 2.8, prove that if $p$is prime and $ab=0$in ${\mathrm{\mathbb{Z}}}_p$, then $a=0$or $b=0$.

Short answer

It is proved that if is prime and $ab=0$in ${\mathrm{\mathbb{Z}}}_n$, then $a=0$or $b=0$.

Step-by-step solution

Use the given part

It is given that $p$is prime and $ab=0$in ${\mathrm{\mathbb{Z}}}_n$.

Proof

As $ab=0$, then for any prime $p|\left(ab\right)$.

Then $p$divides $a$or $b$, that is, $p|a$or $p|b$.

This implies $a=0$or $b=0$.