Question

(a) Solve: 7X ? 1 (mod 19).

(b) Determine [7]^–1 in Z₁₉

Answer 1

a) Given 7x ? 1 (mod 19)

Here gcd(7,19) = 1. Hence the congruence has a unique solution.

Since gcd(7,19) = 1, there exist integers u,v such that 7u+19v = 1.

Here u = -8, v = 3. Therefore 7*(-8)+19*3 = 1 and this implies 7*(-8) ? 1 (mod 19).

Here x = -8 is a solution.

All solutions are x ? -8 (mod 19) ,i.e., x ? 11 (mod 19).

All the solutions are congruent to x ? 11 (mod 19) and therefore the given congruence has a unique solution.

b) [7]-1 = 7-1 = 6.

Therefore, [7]-1 in z₁₉ is 6.