Abstract Algebra: Theory and Applications · Problem 2.4.22

Groups — Thomas Judson

Mathematics — Abstract Algebra · congruence-classes-division-algorithm

Problem

Let

n \in \mathbb{N}

. Use the division algorithm to prove that every integer is congruent mod

n

to precisely one of the integers

0, 1, \ldots, n-1

. Conclude that if

r

is an integer, then there is exactly one

s

\mathbb{Z}

such that

0 \leq s < n

and

[r] = [s]

. Hence, the integers are indeed partitioned by congruence mod

n