# Least Common Multiple

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## least common multiple

[′lēst ′käm·ən ′məl·tə·pəl]
(mathematics)
The least common multiple of a set of quantities (for example, numbers or polynomials) is the smallest quantity divisible by each of them. Abbreviated lcm.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Least Common Multiple

The least common multiple (LCM) of two or several natural numbers is the smallest positive number exactly divisible by each of the given numbers. For example, the LCM of 2 and 3 is 6, and the LCM of 6, 8, 9, 15, and 20 is 360. Least common multiples are used in adding and subtracting fractions; the least common denominator of two or several fractions is the LCM of their denominators. If we know the prime factors of the given numbers, then the LCM of these numbers is the product of all the factors, each taken the greatest number of times it occurs in any one of the numbers. Thus 6 = 2.3, 8 = 2.2.2, 9 = 3.3, 15 = 3–5, and 20 = 2–2–5; therefore, the LCM of 6, 8, 9, 15, and 20 is 2.2.2.3.3.5 = 360. The concept of LCM is applicable not only to numbers; for example, the LCM of two or several polynomials is the polynomial of least degree divisible by each of the given polynomials.

References in periodicals archive ?
Abstract For any positive integer n, let SL(n) denotes the least positive integer k such that L(k) [equivalent to] 0 (mod n), where L(k) denotes the Least Common Multiple of all integers from 1 to k.
For any positive integer n, the famous F.Smarandache LCM function SL(n) defined as the smallest positive integer k such that n | [1, 2, ..., k], where [1, 2, ..., k] denotes the least common multiple of 1, 2, ..., k.
Let L(n) denotes the least common multiple of all positive integers from 1 to n, then we have
For any positive integer n, the famous F.Smarandache LCM function SL(n) is defined as the smallest positive integer k such that n j [1, 2,..., k], where [1, 2,..., k] denotes the least common multiple of all positive integers from 1 to k.
Murthy [3] proved that if n be a prime, then SL(n) = S(n), where SL(n) defined as the smallest positive integer k such that n | [1, 2,..., k], and [1, 2,..., k] denotes the least common multiple of 1, 2,..., k.

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