Modular Inverse

A modular inverse of an integer b (modulo m) is the integer b^(-1) such that

 bb^(-1)=1 (mod m).

A modular inverse can be computed in the Wolfram Language 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. In fact, a nonzero integer b has a modular inverse modulo m iff b and m are relatively prime. For example, 1^(-1)=1 (mod 4) and 3^(-1)=3 (mod 4), but 2 does not have a modular inverse.


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

If b and m are relatively prime, there exist integers x and y such that bx+my=1, and such integers may be found using the Euclidean algorithm. Considering this equation modulo m, it follows that bx=1; i.e., x=b^(-1) (mod m).

If b and m are relatively prime, then Euler's totient theorem states that b^(phi(m))=1 (mod m), where phi(m) is the totient function. Hence, b^(-1)=b^(phi(m)-1) (mod m).

See also

Congruence, Congruence Equation, Linear Congruence Equation

Portions of this entry contributed by Nick Hobson (author's link)

Portions of this entry contributed by Reid Nichol

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Sloane, N. J. A. Sequence A102057 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Modular Inverse

Cite this as:

Hobson, Nick; Nichol, Reid; and Weisstein, Eric W. "Modular Inverse." From MathWorld--A Wolfram Web Resource.

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