Help on understanding this congruency

Question

It is really simple but somehow I cannot connect the dots.

If $p$ is an odd prime, how come $-1 \not\equiv 1 \pmod p$ ?

Answer 1

because if $p$ is an odd prime then it has the form of $p = 2k+1$ for some positive integer $k$ and $-1 \equiv 1 \pmod{n}$ requires that $n \mid -1 -1=-2$

because $$a \equiv b \pmod n \iff n \mid a -b$$ however, $p = 2k+1$ and surely, $2k + 1 \not \mid -2$ because no odd number greater than 1 divides $2$ or $-2$ and so $-1 \not\equiv 1 \pmod{p}$

Generally , if $p > 2$ it will work also.