Inverse Modulo Calculator

Modular arithmetic operates within a finite set of integers that wrap around after reaching a specific value known as the modulus. Depending on the context, an inverse can be either additive or multiplicative. An additive inverse of an integer a modulo m is a number x such that (a + x) ≡ 0 (mod m). This is straightforwardly found using x = (m - (a mod m)) mod m. A multiplicative inverse, which is far more common in advanced applications like RSA encryption, requires that (a × x) ≡ 1 (mod m). A multiplicative inverse only exists if a and m are coprime, meaning their greatest common divisor (gcd) is exactly 1. When gcd(a, m) = 1, the Extended Euclidean Algorithm is deployed to compute coefficients that satisfy Bézout's identity, yielding the precise inverse.

Let us find the multiplicative inverse of integer a = 3 modulo m = 11. First, we confirm that 3 and 11 are coprime by checking their greatest common divisor: gcd(3, 11) = 1. Because the gcd equals 1, a modular inverse is guaranteed to exist. Next, we search for an integer x such that (3 × x) ≡ 1 (mod 11). By testing integer values or applying the Extended Euclidean Algorithm, we find that when x = 4, the product is 3 × 4 = 12. Dividing 12 by 11 leaves a remainder of 1, because 12 mod 11 = 1. Thus, the solution number is 4, yielding a product multiplicative value of 12 and a gcd number of 1.

Keep these best practices in mind when working with modular inverses: Always verify that your integer and modulus are coprime before attempting to find a multiplicative inverse; if their greatest common divisor is greater than 1, no inverse exists. Remember that modular inverses are not unique integers, but rather equivalence classes. Any integer congruent to your result modulo m is also a valid solution, though standard practice is to report the unique integer residing in the range from 0 to m - 1.