If x is congruent to -1 mod p , then how do we prove that x is congruent to $p$-$1$ mod $p$

Question

If x is congruent to -1 mod p , then how do we prove that x is congruent to $p$-$1$ mod $p$ . What i did as per definition of congruence $x$+$1$=$p$*$r$ as according to congruence definition x is congruent to $a$ mod $b$ ,if $x$-$a$ is divisible by $b$ .

Answer 1

Just use the definition, $a \equiv b \pmod m \iff m|a-b.$

So, $x \equiv -1 \pmod p \iff p|(x+1)\implies x+1=pk\space (k \in \Bbb Z$)

$\therefore x+1-p= pk-p=p(k-1)$, which is clearly a multiple of $p$.

So, we get that $p|(x+1-p) \implies p|\{x-(p-1)\} \iff x \equiv p-1 \pmod p $.