This document discusses properties of congruences. It states that if the greatest common divisor (gcd) of a and m is 1, then there exists a multiplicative inverse of a modulo m that can be found in O(log^3 m) bit operations. It also states that if p is a prime number, then every nonzero residue class modulo p has a multiplicative inverse that can be found in O(log^2 p) bit operations. Finally, it introduces solving a linear congruence ax ≡ b (mod m) and states that a solution exists if and only if the gcd of a and m divides b, and in this case the congruence is equivalent to the congruence a'x