Question

Solve for j: k * j ≡ 1 (mod z), where k = 11 and z = 24.

A. j = 2
B. j = 5
C. j = 23
D. j = 14

Answer 1

To solve for j in the equation k * j ≡ 1 (mod z), where k = 11 and z = 24, use the Extended Euclidean Algorithm to find the multiplicative inverse of 11 modulo 24. The multiplicative inverse is 13, so the solution is j = 13.

Step-by-step explanation:

To solve for j in the equation k * j ≡ 1 (mod z), where k = 11 and z = 24, we need to find a value for j that satisfies the congruence equation. We can rewrite the equation as j ≡ (1/k) (mod z). To find the multiplicative inverse of 11 modulo 24, we can use the Extended Euclidean Algorithm.

1. Apply the Extended Euclidean Algorithm to find the GCD of 11 and 24. The algorithm gives us the equation 11 * (-11) + 24 * 5 = 1.
2. Take the coefficient of 11, which is -11, as the multiplicative inverse of 11 modulo 24.
3. Reduce the coefficient modulo 24 to obtain a positive value. In this case, -11 + 24 = 13, which is the multiplicative inverse of 11 modulo 24.

Therefore, j = 13 is the solution to the equation k * j ≡ 1 (mod z).