# For any integers $a$ and $b$, $ab = 0$ implies $a = 0$ or $b = 0$. Prove that this remains true mod prime numbers but not true mod a composite number.

I roughly understand modular arithmetic but I am having trouble starting the problem. I can prove it for just integers but I can't seem to relate it to mod primes and composites?

• By definition, a composite number $q$ is the product of two smaller integers. What is that product modulo $q$? – eyeballfrog Sep 27 at 7:15

1. Start with a single example. Is the statement true in the integers modulo $$6$$? Why not, exactly? How does your argument generalize to other composite numbers?
2. Start by writing out definitions. What does it mean for a modulus $$n$$ to be a composite number? What does it mean for an integer to be congruent to $$0$$ modulo $$n$$? Can you find a way to plug these definitions into the problem you're trying to solve?
Well, in the residue class ring $${\Bbb Z}_m$$, $$m\geq 2$$, each element $$a\ne 0$$ is either a unit or a zero divisor. An element $$a\ne 0$$ is a unit if $$\gcd(a,m)=1$$, otherwise a zero divisor.
The implication $$ab=0\rightarrow a=0\vee b=0$$ holds iff $$a$$ or $$b$$ is not a zero divisor.