Divisibilités

Divisibilités, in mathematics, refers to the concept of whether one integer can be divided by another integer without leaving a remainder. When an integer 'a' can be divided by an integer 'b' such that the result is also an integer, 'a' is said to be divisible by 'b'. Conversely, 'b' is said to be a divisor of 'a'.

This relationship is fundamental in number theory. For instance, the number 12 is divisible by 1, 2, 3, 4, 6, and 12. These numbers are the divisors of 12. The number 0 is divisible by any non-zero integer, as any integer multiplied by 0 results in 0. However, division by zero is undefined.

The concept of divisibility is often expressed using modular arithmetic. An integer 'a' is divisible by an integer 'b' if and only if 'a' modulo 'b' is equal to 0. This is written as a ≡ 0 (mod b).

Divisibility rules are shortcuts used to determine if a number is divisible by certain smaller integers without performing the actual division. For example, a number is divisible by 2 if its last digit is even, and a number is divisible by 5 if its last digit is 0 or 5.

Understanding divisibility is crucial for various mathematical operations and concepts, including prime factorization, greatest common divisors (GCD), and least common multiples (LCM). It forms the bedrock of much of elementary arithmetic and number theory.