integermultiple

An integer multiple of an integer b is any integer a for which there exists an integer k with a = k × b. The set of all integer multiples of b is {..., -2b, -b, 0, b, 2b, 3b, ...}, an infinite arithmetic progression with common difference b. The concept is closely tied to divisibility: a is an integer multiple of b if and only if b divides a. For example, 12 is an integer multiple of 3 (12 = 4 × 3), and 0 is a multiple of every integer. Negative multiples are also included, such as -15 being a multiple of 5 (-3 × 5).

Common multiples and the least common multiple (LCM) are standard notions related to integer multiples. Two integers a and b have common multiples, and the smallest positive one is called their least common multiple. The set of common multiples of a and b is precisely the multiples of lcm(a, b); for example, the common multiples of 6 and 8 are all multiples of 24: ..., -48, -24, 0, 24, 48, ....

Generalizations extend the idea beyond integers. In abstract algebra, an element a is an integer multiple of b if there exists an integer n with a = n·b, within a given ring or module. In the integers this matches the divisibility relation. Thus, the concept of an integer multiple provides a simple way to express how one number scales another by an integer factor and to discuss related notions such as divisibility and least common multiples.