 TOPICS # Modular Inverse

A modular inverse of an integer (modulo ) is the integer such that A modular inverse can be computed in the Wolfram Language using PowerMod[b, -1, m].

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

If is not prime, then not every nonzero integer has a modular inverse. In fact, a nonzero integer has a modular inverse modulo iff and are relatively prime. For example, (mod 4) and (mod 4), but 2 does not have a modular inverse. The triangle above (OEIS A102057) gives modular inverses of (mod ) for , 2, ..., and , 3, .... 0 indicates that no modular inverse exists.

If and are relatively prime, there exist integers and such that , and such integers may be found using the Euclidean algorithm. Considering this equation modulo , it follows that ; i.e., .

If and are relatively prime, then Euler's totient theorem states that , where is the totient function. Hence, .

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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## References

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

Modular Inverse

## Cite this as:

Hobson, Nick; Nichol, Reid; and Weisstein, Eric W. "Modular Inverse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ModularInverse.html