Prove that if A, a non zero element in Zn (integers mod n) is not a unit then A is a zero divisor in Zn.
The Attempt at a Solution
[AB] does not equal 1 mod n for some A and all B in Zn
=> the element A is a multiple of n so n divides A because otherwise there should be a remainder of 1.
=> AB=0 for all B in Zn
So A is a zero divisor in Zn.
I feel the first implication is a bit unrigorous.