a is congruent to b modulo m denoted as a \equiv b (mod \, n) , if a and b have the remainder when they are divided by n, for a, b \in \mathbb{Z}. Two integers a ...a is congruent to b modulo m denoted as a \equiv b (mod \, n) , if a and b have the remainder when they are divided by n, for a, b \in \mathbb{Z}. Two integers a and b are said to be congruent modulo n, a \equiv b (mod \, n), if all of the following are true: Proof: Let a, b \in \mathbb{Z} such that a \equiv b (mod n). Proof: Let a, b, c \in\mathbb{Z}, such that a \equiv b (mod n) and b \equiv c (mod n).