"ab = 0" iff "a = 0 or b = 0" for all real a, b.

i.e. if a product for two numbers is zero, then at least one of them must be zero. Likewise, like Roy mentioned, any product with a zero gives you zero.

Same reason cannot hold for a product that is non-zero because there are infinite ways to give a non-zero product involving real numbers.

There is a finite number of ways to give you a non-zero product IF you are restricted to only integer solutions though:

For example, ab = -8 iff

a = -8, b = 1 or

a = -4, b = 2 or

a = -2, b = 4 or

a = -1, b = 8 or

a = 1, b = -8 or

a = 2, b = -4 or

a = 4, b = -2 or

a = 8, b = -1.

If none of these combinations solve the given equation then we can say that the equation has no integer solution.