Multiplicative Inverse Calculator

A multiplicative inverse is a value that reverses multiplication. In ordinary arithmetic, the inverse of a nonzero number a is its reciprocal, 1 divided by a, because multiplying a by 1 divided by a gives 1. The reciprocal of 8 is 0.125, the reciprocal of one half is 2, and the reciprocal of a negative number is also negative. Zero has no multiplicative inverse because no finite value multiplied by zero can produce 1. Modular arithmetic uses a related but distinct definition. The modular inverse of an integer a modulo m is an integer x for which a times x leaves remainder 1 when divided by m. For example, 7 has inverse 15 modulo 26 because 7 times 15 is 105 and 105 mod 26 is 1. Modular inverses are not generally the same as decimal reciprocals. They belong to a finite system of residue classes, and adding or subtracting any multiple of the modulus gives another representative of the same modular answer. A modular inverse exists exactly when the number and modulus are coprime, meaning their greatest common divisor is 1. The calculator checks this condition with the Extended Euclidean Algorithm. That algorithm finds coefficients connecting the inputs to their greatest common divisor. When the greatest common divisor is 1, the coefficient attached to the number can be reduced modulo m to obtain the least nonnegative inverse. If the inputs share a factor greater than 1, no multiplication can produce remainder 1, so the calculator reports that the requested inverse does not exist. Choose the reciprocal mode for fractions, scaling, algebraic division, rates, and real-number calculations. Choose modular inverse for congruences, number theory, cryptography, checksums, and integer algorithms. Modular inputs must be safe integers and the modulus must be greater than 1. Negative integers are accepted as the number because they can be reduced to an equivalent residue before the inverse is interpreted. Multiplicative inverses are central to solving equations. Multiplying both sides of an ordinary equation by a reciprocal isolates a nonzero coefficient. In a modular equation, multiplying by a modular inverse isolates a variable only when that inverse exists. Public-key cryptography uses this idea with much larger integers, although production cryptographic work requires specialized arbitrary-precision and constant-time software. This calculator is intended for learning, checking homework, and solving moderate-sized arithmetic examples with a clear distinction between the two inverse concepts.