Proof of zero divisor existence.

[shamus390]

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

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

 

[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 $\phi$?