Proof of zero divisor existence.

1. Feb 22, 2013

shamus390

1. Let a != 0 and b be elements of the integers mod n. If the equation ax=b has no solution in Zn then a is a zero divisor in Zn

3. The attempt at a solution

Not sure where to start on this proof, I keep trying to find something using the properties of modular arithmetic but am coming up empty

2. Feb 22, 2013

jbunniii

Hint: If $ax = b$ has no solutions, then that means the map $\phi : Z_n \rightarrow Z_n$ defined by $\phi(x) = ax$ is not surjective. Since $Z_n$ is finite, what else does that imply about $\phi$?