I answered a question about whether zero is prime or composite on Khan Academy a while ago. Since then, two people have commented on my answer, asking another question that I don't know the answer to.
Are there zero divisors (numbers such that, when multiplied by some nonzero number, the product is zero) other than zero? If so, what are they? Wouldn't that go against the Zero Product Property, which states that if you take two numbers, $n$ and $m$, and multiply them together to get zero, then either $n$ or $m$ is equal to zero? I.e. if $nm=0$, then either $n=0$ and $m=0$.