Fact: If a and b are two integers, then ab = 0 if and only if a = 0 or b = 0.
Proof: If a = 0, then
We can add the negative of (0)b to both sides and get
If b = 0, then since multiplication is commutative, we could relabel a and b and use the last argument
For the converse, let us assume that neither a nor b is 0. We will consider the possible cases.
If a and b are both positive integers, then b is greater than or equal to 1 so ab is greater than or equal to a which is greater than 0. We conclude that ab is not zero.
In the other cases, the absolute value of the product is the product of the absolute values, and will, hence, not be 0.