15 has no primitive root as the orders of 2, 4, 7, 8, 11, 13, and 14 modulo 15 are 4, 2, 4, 2, 4, 2, and 2, respectively, and none equal φ(15) = 8.
To determine whether 15 has a primitive root, we calculate the orders of potential primitive roots modulo 15. The orders are the smallest positive integers 'k' such that ak ≡1(mod15). We examine the orders of numbers 2, 4, 7, 8, 11, 13, and 14 modulo 15.
For 2, 4, 7, 8, 11, 13, and 14, the respective orders are 4, 2, 4, 2, 4, 2, and 2. None of these orders equals the totient function value, φ(15)=8. Since a primitive root modulo 'm' must have an order equal to φ(m), and none of the tested numbers satisfies this condition for 15, we conclude that 15 has no primitive root. The calculated orders provide insights into the cyclic behavior of residues modulo 15, demonstrating the absence of a primitive root among the tested candidates.