Problem 5 Let p be an odd prime and suppose that a and b are

Problem 5. Let p be an odd prime, and suppose that a and b are primitive roots modulo p. Prove that ab is not a primitive root modulo p.

Solution

given that ab are premitive roots modulo p

We then have 1 = 1ep(a) (ab) ep(a) (mod p) = a ep(a) b ep(a) (mod p) b ep(a) (mod p).

But since order of b modulo p has to divide any power of b that gives 1 modulo p, we have ep(b)|ep(a). On the other hand, we could have done this by starting with (ab) ep(b) to deduce that ep(a)|ep(b). It then must follow that ep(a) = ep(b).

now ab are not a primitive roots modulo p