Multiplicative Inverse Modulo Calculator

In modular arithmetic, the multiplicative inverse of an integer A modulo M is an integer X such that their product is congruent to 1 modulo M. Mathematically, this is expressed as (A * X) ≡ 1 (mod M). For this inverse to exist, A and M must be coprime, meaning their greatest common divisor must equal 1. The calculator determines this by evaluating the greatest common divisor using my_gcd = greatest_comm_div(a,m) and checking if its absolute value (gcd_real) equals 1. If valid, the extended Euclidean algorithm computes the raw inverse (res_temp = inverse_modulo(a,m)), normalizes it into a positive range using res = max(res_temp, res_temp + m), and verifies the product via ares = a * res.

Let us find the multiplicative inverse of A = 3 modulo M = 11. First, we confirm that 3 and 11 are coprime because their greatest common divisor is 1 (gcd_real = 1). Next, applying the extended Euclidean algorithm, we find the base inverse solution res_temp = -7. To ensure our result falls within the standard positive range from 0 to M-1, we apply our normalization rule: res = max(-7, -7 + 11), which results in res = 4. We can verify this calculation using the product formula ares = 3 * 4 = 12. Since 12 modulo 11 leaves a remainder of 1, our computed modular inverse of 4 is verified as correct.

Always verify that your number A and modulus M share no common factors other than 1 before attempting to find an inverse; otherwise, no solution exists. Remember that modular inverses are always expressed within the range of 0 to M-1. Double-check your arithmetic when using manual methods like the Extended Euclidean Algorithm to avoid sign errors with negative intermediate remainders.