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.

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 ?
In all this work, we consider monoids M with a finite generating set S satisfying the following properties: M is atomic, left-cancellative (if a, u, v [member of] M are such that au = av, then u = v) and verifies that if a subset of S has a right common multiple, then it has a least right common multiple.
For such a monoid, if J [subset] S is such that J has a common multiple, then a least common multiple (lcm) exists and is unique.
We will call cliques the subsets of S having a common multiple, and let J be the set of all cliques; if J is a clique, we note [M.
Eventually, several rhythmic strategies emerged: (1) solving polyrhythms by means of calculating the least common multiple of their constituent components, (2) translating rhythmic notations into indications of tempo, and, (3) casting one line of a polyrhythm as strongly foreground in nature against which other rhythmic lines act ornamentally in varying degrees of rhythmic dissonance to the original.
In instances where two or more rhythmic lines share both beginning and end points, a method of calculating the least common multiple of the rhythmic components works well.
This grid can be simplified by taking the least common multiple of two rhythms - in this case 60 (Example 2).
This experience is also an ethnomathematical introduction to least common multiple (LCM).
The least common multiple (LCM) of two numbers is the smallest positive number that is a multiple of both.
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.
Let L(n) denotes the least common multiple of all positive integers from 1 to n, then we have