Page 1: Divisibility, Euclidean Division, and Congruence in Z

This page introduces key concepts related to divisibilité dans Z and congruence relations. It covers the definitions of multiples and divisors, properties of divisibility, the Euclidean division theorem, and basic properties of congruences.

The page begins by defining divisibility in Z. For integers a and b, if a divides b, it is denoted as a|b, meaning b = ak for some integer k. This leads to the concepts of multiples and divisors.

Definition: A multiple of a is any integer b such that a|b. Conversely, a is called a divisor of b.

Some important properties of divisibility are highlighted:

1. 0 is a multiple of every integer.
2. Every non-zero integer divides itself and 1.
3. Divisibility is transitive: if a|b and b|c, then a|c.
4. Linear combination property: if a|b and a|c, then a|$mb + nc$ for any integers m and n.

The page then introduces the théorème de la division euclidienne (Euclidean division theorem) for integers:

Definition: For any integer a and positive integer b, there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b.

This theorem is fundamental in number theory and has many applications. The uniqueness of q and r is emphasized, noting that r is the only multiple of b between -b and b.

Example: When dividing by 3, the possible remainders are 0, 1, and 2. Any integer can be written in the form 3k, 3k+1, or 3k+2 for some integer k.

Finally, the page introduces congruence modulo and some of its basic properties:

Definition: Two integers a and b are congruent modulo m if their difference a - b is divisible by m. This is denoted as a ≡ b (mod m).

The properties of congruence with respect to addition and multiplication are stated:

1. If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).
2. If a ≡ b (mod m) and c ≡ d (mod m), then ac ≡ bd (mod m).

These properties are crucial for understanding modular arithmetic and its applications in various areas of mathematics and computer science.

Highlight: The concepts of divisibility and congruence form the foundation for many advanced topics in number theory and abstract algebra.