Why can I cancel in modular arithmetic when working modulus a prime number?

Working modulus a prime number in modular arithmetic let's you cancel factors in a congruence equation. Let p and k be integers, p a prime number and k not a multiple of p:

$a \cdot k\equiv b \cdot k\pmod n$

We can multiply by a constant on each side and maintain the congruence. Let this constant be a multiplicative inverse of k (which is guaranteed to exist in this case).

$a \cdot k \cdot k^{-1}\equiv b \cdot k \cdot k^{-1}\pmod n$

Why is it that I can now justify canceling the initial k? $k \cdot k^{-1}$ gives some integer $m$, which when divided by $n$ gives remainder 1. But what is the property that takes me from $a \cdot m\equiv b \cdot m\pmod n$ to $a\equiv b\pmod n$?