Do there exist integers $m,x,$ and $y$ such that in $\mathbb Z_m$ we have $[x]\neq [0]$ and $[y]\neq [0],$ but $[x][y] = [0]$

Question

I'm working on discrete math, and this problem is stumping me. I feel like it shouldn't be that hard, but I can't wrap my head around it.

Answer 1

You are trying to find two numbers that are not divisible by $m$, but whose product $xy$ is divisible by $m$. So choose $x$ so that $x$ has some, but not all, of the prime factors of $m$, and $y$ has the others. Neither $x$ nor $y$ will be divisible by $m$, since they are each missing some of $m$'s prime factors, but together the product $xy$ has all of the prime factors of $m$.