Proof Fermat's Little Theorem

Question

I'm trying to follow the proof for here https://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html?fbclid=IwAR2WAws38n6M--Dx4cOWc8QToKG9sJf1JL9V_9MDATTG4UIx2yNyvpF2M7Q

I'm a bit confused by the part

Suppose that ra and sa are the same modulo p, then we have $r = s (mod p)$, so the p-1 multiples of a above are distinct and nonzero;

why do we have $r = s\ (mod\ p)$ ?

Answer 1

$p\mid ra-sa=(r-s)a$. Since $p$ is prime and by hypothesis $p\nmid a$, it must hold $p\mid r-s$.

Answer 2

If $p$ is a prime number then $\mathbb{Z}_p$ has a cancellation property, you can see the Saucy's proof. Although, in general it's not true, for example in $\mathbb{Z}_6$: $2\cdot3 = 2\cdot0 =0$ but $3 \not =0$.