Question

Find 245⁻¹ in Z/748 and write the answer as b where 0≤b<748.

Answer 1

To find the multiplicative inverse of 245 in Z/748, we can use the Extended Euclidean Algorithm. By applying the Euclidean Algorithm, we can express 1 as a linear combination of 245 and 748. Solving for the unknowns in this expression gives us the multiplicative inverse of 245 modulo 748, which is 345.

Step-by-step explanation:

To find 245⁻¹ in Z/748, we need to find the multiplicative inverse of 245 modulo 748. The multiplicative inverse of a number x modulo a is a number y such that (x * y) % a = 1. In this case, we need to find y such that (245 * y) % 748 = 1.

1. We can use the Extended Euclidean Algorithm to find the multiplicative inverse. The algorithm states that if gcd(245, 748) = 1, then there exists an integer y such that (245 * y) % 748 = 1.
2. Using the Euclidean Algorithm, we have:

• 748 = 3 * 245 + 13
• 245 = 18 * 13 + 11
• 13 = 1 * 11 + 2
• 11 = 5 * 2 + 1

Working backwards, we can express 1 as a linear combination of 245 and 748:

• 1 = 11 - 5 * 2
• 1 = 11 - 5 * (13 - 1 * 11)
• 1 = 6 * 11 - 5 * 13
• 1 = 6 * (245 - 18 * 13) - 5 * 13
• 1 = 6 * 245 - 113 * 13
• 1 = 6 * 245 - 113 * (748 - 3 * 245)
• 1 = 345 * 245 - 113 * 748

Hence, the multiplicative inverse of 245 modulo 748 is 345. So, 245⁻¹ ≡ 345 (mod 748).