The set {1, 2, 4, 7, 8, 11, 13, 14} is a group under multiplication modulo 15. The inverses of 4 and 7 are respectively:...

Question

The set {1, 2, 4, 7, 8, 11, 13, 14} is a group under multiplication modulo 15. The inverses of 4 and 7 are respectively:

A. 3 and 13
B. 2 and 11
C. 4 and 13
D. 8 and 14

Accepted Answer

C. 4 and 13 In the multiplicative group Z_15^* (integers coprime to 15), the inverse of an element a is the element b such that a*b ≡ 1 (mod 15). To find the inverse of 4, multiply candidates until the product is 1 mod 15; similarly for 7. Here 4*4 = 16 ≡ 1 (mod 15), so 4 is its own inverse, and 7*13 = 91 = 6*15+1 ≡ 1 (mod 15), so inverse of 7 is 13.

The set {1, 2, 4, 7, 8, 11, 13, 14} is a group under multiplication modulo 15. The inverses of 4 and 7 are respectively: A. 3 and 13 B. 2 and 11 C. 4 and 13 D. 8 and 14