3.41. Let p be a prime satisfying p = 1 (mod 3). We say that a is a cubic residue modulo p if pa and there is an integer c satisfying a = c³ (mod p). (a) Let a and b be cubic residues modulo p. Prove that ab is a cubic residue mod- ulo p. (b) Give an example to show that (unlike the case with quadratic residues) it is possible for none of a, b, and ab to be a cubic residue modulo p. (c) Let g be a primitive root modulo p. Prove that a is a cubic residue modulo p if and only if 3 | log, (a), where log, (a) is the discrete logarithm of a to the base g. (d) Suppose instead that p = 2 (mod 3). Prove that for every integer a there is an integer c satisfying a c³ (mod p). In other words, if p = 2 (mod 3), show that every number is a cube modulo p.