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For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and −2 as well.
Here, 36 is the least common multiple of 12 and 18. Their product, 216, is also a common denominator, but calculating with that denominator involves larger numbers:
lcm(m, n) (least common multiple of m and n) is the product of all prime factors of m or n (with the largest multiplicity for m or n). gcd(m, n) × lcm(m, n) = m × n. Finding the prime factors is often harder than computing gcd and lcm using other algorithms which do not require known prime factorization.
14, 49, −21 and 0 are multiples of 7, whereas 3 and −6 are not. This is because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and −21, while there are no such integers for 3 and −6.
LCM may refer to: Computing and mathematics. Latent class model, a concept in statistics; Least common multiple, a function of two integers; Living Computer Museum;
gcd(a, b) is closely related to the least common multiple lcm(a, b): we have gcd(a, b)⋅lcm(a, b) = | a⋅b |. This formula is often used to compute least common multiples: one first computes the GCD with Euclid's algorithm and then divides the product of the given numbers by their GCD. The following versions of distributivity hold true:
The least common multiple of a and b is equal to their product ab, i.e. lcm(a, b) = ab. [4] As a consequence of the third point, if a and b are coprime and br ≡ bs (mod a), then r ≡ s (mod a). [5] That is, we may "divide by b" when working modulo a.
Visualisation of using the binary GCD algorithm to find the greatest common divisor (GCD) of 36 and 24. Thus, the GCD is 2 2 × 3 = 12.. The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, [1] [2] is an algorithm that computes the greatest common divisor (GCD) of two nonnegative integers.