Let p be an odd prime. If c is a quadratic residue mod p, prove that there are only two solutions for the ....?

(a) Since c is a quadratic residue mod p, n^2 = c mod p for some integer n.

Thus, x^2 = c = n^2 mod p
==> x^2 - n^2 = 0 mod p
==> (x + n)(x - n) = 0 mod p.
Since p is a prime, Z_p is a field and thus we conclude that
x + n = 0 mod p or x - n = 0 mod p.
==> x = n or -n mod p, yielding the two solutions (since n is not 0).

(Alternately, Lagrange's Theorem tells us there are no more than two solutions, since p is prime.)

(b) Look at x^2 = 1 mod 8. This has four solutions mod 8:
x = 1,3,5,7.

I hope that helps!