This chapter introduces the basic properties of congruences modulo n, along with the related notion of congruence classes modulo n. Other items discussed include the Chinese remainder theorem, Euler's phi function, arithmetic functions and Möbius inversion, and Fermat's little theorem.

Definitions and basic properties

For positive integer n, and for a, b ∈ ℤ, we say that a is congruent tobmodulon if n | (a - b), and we write a ≡ b (mod n). If n ∣ (a - b), then we write a ≢ b (mod n). The relation a ≡ b (mod n) is called a congruence relation, or simply, a congruence. The number n appearing in such congruences is called the modulus of the congruence. This usage of the “mod” notation as part of a congruence is not to be confused with the “mod” operation introduced in §1.1.

A simple observation is that a ≡ b (mod n) if and only if there exists an integer c such that a = b + cn. From this, and Theorem 1.4, the following is immediate:

Theorem 2.1.Let n be a positive integer. For every integer a, there exists a unique integer b such that a ≡ b (mod n) and 0 ≤ b < n, namely, b ≔ a mod n.