Hi. I'm having trouble with these problems.
p is and odd prime for both.
Prove x = 0 mod p if x = -x mod p.
Prove x = +/-y mod p^2 if x^2 = y^2 mod p^2, neither x nor y are 0 mod p.
Thanks...
Suppose $\displaystyle x>0$, and $\displaystyle x \not\equiv 0 \mod p$, then there exist $\displaystyle k \ge 0$ and $\displaystyle p>r>0$ such that:
$\displaystyle x=kp+r$
Also:
$\displaystyle -x=(-k)p-r=(-1-k)p+(p-r)$
So if $\displaystyle x \equiv -x \mod p$ then $\displaystyle p-r=r$, or $\displaystyle 2r=p$, but $\displaystyle p$ is an odd prime which is a contradiction, so our premis fails and $\displaystyle x \equiv 0 \mod p$.
RonL