The question asks you to find if a statement is true, if it is, prove it. If it is not, give a counterexample.
If a2 = b2 (mod p), then a = b (mod p), where p is prime.
Note the underlined equal signs are actually meant to be congruence signs. Is there a way to insert congruence signs in this forum?
I based my proof on the properties of congruences which states that if a = b (mod m), and c = d (mod m), then ac = bd (mod m).
a = b (mod m)
a = b + mk
(a)(a) = (b + mk)(b + mk)
a2 = b2 + 2mbk + m2k2
a2 = b2 + m(2bk + mk2) 2bk + mk2 is an integer, say t
a2 = b2 + mt
Therefore a2 = b2 (mod m)
I personally don't see why this proof would not hold if the modulus is prime. Then again, I'm not great with proofs. Can I stick a "p" in place of "m" in my proof and call it a day? Or does the fact the modulus is prime complicate things?