Linear Congruences

Congruences in the form ax b (mod m) are called linear congruences.

An important special case is that of multiplicative inverses: what value should a be to satisfy aa 1 (mod m)? Any value for a that satisfies this equation is called a multiplicative inverse of a modulo m.

If we know a is the inverse of 2 modulo 7 then solving 2x 3 (mod 7) can be solved by multiplying both sides of the equation by a. This is possible because of the following theorem:

Theorem: For all integers a, b and c and natural numbers m, If a b (mod m) then ac bc (mod m).