# Why there is no zero-divisors modulo a prime.

Let us say that an integer $k$ where $0< k < m$ is a zero-divisor mod $m$ if $kn \equiv 0 \pmod{m}$ for some $n$ with $0 < n < m$.

Prove the following: If $m$ is prime then no integer $k$ is a zero-divisor mod $m$.

Homework question; I think I'm on my way to the solution?

So $kn/m$ needs to be an integer. Since $k < m$ and $n < m$, we know that neither $k$ nor $m$ is equal to $m$. Furthermore, $k > 0$ and $n > 0$, therefore $k n$ is not equal to zero. Thus $k n > 0$ and $k n$ is not divisible by $m$ (as $m$ is a prime) and therefore, there exists no integer $k$ that is a zero-divisor mod $m$.

Is that a valid proof for the question at hand? If not, what is?

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This is an immediate corollary of your prior question, that prime $\,m\mid kn\,\Rightarrow\ m\mid k\,$ or $\,m\mid n.\ \$ –  Bill Dubuque May 2 '14 at 18:45
I think I'm misunderstanding the question.. Are we given the fact that m | kn? Or do we have to make that conclusion on our own? Furthermore, your solution is similar to that of the book's, however, I'm not really sure why that statement would prove the question at hand.. Can you explain? –  user122661 May 2 '14 at 18:52
Yes, $\,kn\equiv 0\pmod m\iff m\mid kn.\$ Now apply the prior result. –  Bill Dubuque May 2 '14 at 19:00
Okay, so I now know m|k OR m|n but WHY does this imply that that no integer k is a zero-divisor mod m? –  user122661 May 2 '14 at 19:02
That contradicts $\,0 < k,n < m.\ \$ –  Bill Dubuque May 2 '14 at 19:55