Portions of this entry contributed by Reid Nichol

A modular inverse of an integer b (modulo m) is the integer such that

A modular inverse can be computed in Mathematica using PowerMod[b, -1, m].

Every nonzero integer b has an inverse (modulo p) for p a prime and b not a multiple of p. For example, the modular inverses of 1, 2, 3, and 4 (mod 5) are 1, 3, 2, and 4.

If m is not prime, then not every nonzero integer b has a modular inverse. For example, (mod 4) and (mod 4), but 2 does not have a modular inverse.

The triangle above (Sloane's A102057) gives modular inverses of b (mod m) for b = 1, 2, ..., and m = 2, 3, .... 0 indicates that no modular inverse exists.

Congruence, Congruence Equation, Linear Congruence Equation

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