Question

Give an example to show that a
2 ≡ b
2
(mod n) need not imply that a ≡
b(mod n)

Answer 1

assuming you meant a/2 ≡ b/2 (mod n)
that means
b/2 = kn + a/2
b = 2kn + a
so b-a = 2kn ≡ 0 (mod n)
Thus a≡b (mod n)