The inverse of a number a modulo m is a number x such that . It exists (and is unique if exists) if and only a and m are relatively prime (that is, gcd(a,m) = 1). In particular, if m is a prime, every non-zero element of Zm has an inverse (thus making it an algebraic structure known as field).
Conventionally, the mathematical notation used for inverses is .
In modular arithmetic the inverse of a is analogous to the number 1 / a in usual real-number arithmetic. If you have a product c = ab, and one of the factors has an inverse, you can get the other factor by multiplying the product by that inverse: . Thus you can perform division in ring Zm.
 Finding the inverse
We can rewrite the defining equation of modular inverses as an equivalent linear diophantine equation: ax + my = 1. This equation has a solution whenever gcd(a,m) = 1, and we can find such solution (x,y) by means of the extended Euclidean algorithm.
Then , and also .
The following Python code implements this algorithm.
# Iterative Algorithm (xgcd) def iterative_egcd(a, b): x,y, u,v = 0,1, 1,0 while a != 0: q,r = b//a,b%a; m,n = x-u*q,y-v*q # use x//y for floor "floor division" b,a, x,y, u,v = a,r, u,v, m,n return b, x, y # Recursive Algorithm def recursive_egcd(a, b): """Returns a triple (g, x, y), such that ax + by = g = gcd(a,b). Assumes a, b >= 0, and that at least one of them is > 0. Bounds on output values: |x|, |y| <= max(a, b).""" if a == 0: return (b, 0, 1) else: g, y, x = recursive_egcd(b % a, a) return (g, x - (b // a) * y, y) egcd = iterative_egcd # or recursive_egcd(a, m) def modinv(a, m): g, x, y = egcd(a, m) if g != 1: return None else: return x % m
 Alternative algorithm
If you happen to know φ(m), you can also compute the inverses using Euler's theorem, which states that . By multiplying both sides of this equation by a's modular inverse, we can deduce that: .
And so you can utilize repeated squaring algorithm to quickly find the inverse.
This algorithm can be useful if m is a fixed number in your program (so, you can hardcode a precomputed value of φ(m)), or if m is a prime number, in which case φ(m) = m - 1. In general case, however, computing φ(m) is equivalent to factoring, which is a hard problem, so prefer using the extended GCD algorithm.
Suppose we need to calculate . If b and p are co-primes (or if one of them is a prime), then we can calculate the modular inverse b' of b.