Find an inverse of 𝑎 modulo 𝑚.a) 𝑎 = 2, 𝑚 = 17

To find the inverse of 𝑎 modulo 𝑚, we need to find a number 𝑏 such that 𝑎𝑏 ≡ 1 (mod 𝑚).

In this case, 𝑎 = 2 and 𝑚 = 17.

Step 1: Start by checking if 𝑎 and 𝑚 are coprime (i.e., their greatest common divisor is 1). In this case, the greatest common divisor of 2 and 17 is 1, so they are coprime.

Step 2: Use the Extended Euclidean Algorithm to find the inverse. The algorithm involves finding two numbers 𝑥 and 𝑦 such that 𝑎𝑥 + 𝑚𝑦 = 1.

Using the Extended Euclidean Algorithm, we have: 17 = 2 * 8 + 1 2 = 1 * 2 + 0

Step 3: Rewriting the equation from Step 2, we have: 1 = 17 - 2 * 8

Step 4: Now, we can see that 𝑏 = -8 is the inverse of 𝑎 = 2 modulo 𝑚 = 17, since 2 * -8 ≡ 1 (mod 17).

Therefore, the inverse of 2 modulo 17 is -8.