Modular arithmetic congruence class simple proof I have the following question but I'm unsure of how it can be approached by a method of proof. I'm new to modular arithmetic and any information on how to solve this would be great for me.

(b) Let $t,s\in\{0,1,2,3,4,5\}$. In $\mathbb Z_{25}$, prove that $[t]\,[s]\neq[24]$.


First note that $24\equiv -1\pmod{25}$ and hence we are trying to show that $ts\not\equiv-1\pmod{25}$.  Suppose for contradiction that $ts\equiv -1\pmod{25}$, then multiplying through by $-1$ we get $-ts\equiv 1\pmod{25}$ so $t$ or $s$ is invertible, say $t$ with inverse $-s$.  Therefore $\gcd(t,25) = 1$  (why?), and hence $t = 1,2,3,4$.  There is a unique value of $s$ corresponding to each of these (why?), each of which should give you a contradiction.
