Question

If $a\ne 0$and $b$are elements of ${\mathrm{\mathbb{Z}}}_n$, and $ax=b$has no solutions in ${\mathrm{\mathbb{Z}}}_n$, prove that $a$is a zero divisor.

Short answer

It is proved that $ax=b$has some solution.

Step-by-step solution

Use the given part

It is given that the equation $ax=b$has no solutions in ${\mathrm{\mathbb{Z}}}_n$.Then $a$is a zero divisor. By using implication, it can be said that if $a$is not a zero divisor then the equation $ax=b$has some solution.

Proof

It is known that each non-zero element of ${\mathrm{\mathbb{Z}}}_n$is either a zero divisor or a unit.

By implication, is not a zero divisor, then it will be a unit, which implies that the equation $ax=b$has some solution.

Hence proved.