If $n =p$ is a prime number and if $\neq[a]$ is in $J_p$ , then there is an element $[b]$ in $J_p$ such that $[a][b] = $. ($J_p$ being the set of congruence classes mod $p$)
This was a remark in the book Topics of Algebra by I.N. Herstein in the congruence modulo section of the topic Integers. He didn't prove it.
I tried to prove it but I have no clue as to how I'm supposed to approach this, all I could do is simplify this to : Prove, for all $a<p$ there exists a $b<p$ such that $ab=np + 1$ where $p$ is prime and $n (<a,b)$ is any integer.
Could anyone show how this is true or maybe help me out with how to approach it.