Multiples

Multiples are numbers obtained by multiplying a given integer by another integer. Formally, the multiples of an integer a are all numbers of the form a × k, where k is an integer. This holds for any a, including zero. If a ≠ 0, the set of multiples includes positive, negative, and zero multiples (for example, multiples of 3 are …, -6, -3, 0, 3, 6, 9, …). If a = 0, the only multiple is 0.

A number b is a multiple of a when a divides b; equivalently, b = a × k for some integer k. This relation defines divisibility: a divides b if and only if b is a multiple of a. The concept contrasts with factors, which are numbers that divide a given number.

Common multiples of two integers a and b are numbers that are multiples of both a and b. The smallest positive common multiple is called the least common multiple (LCM). A useful identity for nonzero integers is LCM(a, b) × GCD(a, b) = |a × b|, linking common multiples to greatest common divisors.

Multiples form an arithmetic progression with common difference a, and the sum of multiples of a is itself a multiple of a. In modular arithmetic, numbers that are congruent to 0 modulo a are exactly the multiples of a.

Applications include simplifying fractions, solving problems involving shared timings or alignments, and foundational ideas in number theory, such as divisibility, LCM computation, and modular reasoning.