Least Common Multiple Calculator

What is the Least Common Multiple (LCM)?

A multiple of a number is the result of multiplying it by a whole number: the multiples of 4 are 4, 8, 12, 16, 20 and so on. The least common multiple of two or more numbers is the smallest positive number that is a multiple of every one of them. The LCM of 4 and 6 is 12. It is also called the lowest common multiple or smallest common multiple, and when the numbers are denominators, the least common denominator.

Brute Force Method

The most direct approach is to list multiples of each number until the same value appears in every list. For 4 and 6, the multiples of 4 are 4, 8, 12, 16 … and the multiples of 6 are 6, 12, 18 …, so the first shared value, 12, is the LCM. A small shortcut is to list only the multiples of the largest number and test each one against the others. This works well for small numbers but becomes tedious quickly when the LCM is large.

Prime Factorization Method

Factor each number into primes. The LCM is the product of every prime that appears in any of the factorizations, each raised to the highest power it reaches in any single number. Where the GCF takes the lowest shared powers, the LCM takes the highest powers of everything.

For the default numbers: 330 = 2 × 3 × 5 × 11, 75 = 3 × 5², 450 = 2 × 3² × 5² and 225 = 3² × 5². The primes involved are 2, 3, 5 and 11. Their highest powers are 2¹, 3², 5² and 11¹, so the LCM is 2 × 9 × 25 × 11 = 4,950. You can check that each of the four numbers divides 4,950 evenly.

Greatest Common Divisor Method

For two numbers there is a neat relationship: the product of the numbers equals the product of their GCF and their LCM. So LCM(a, b) = a × b ÷ GCF(a, b). Because the GCF can be found very quickly with the Euclidean algorithm, this is usually the fastest way to compute an LCM, especially for large numbers where factoring would be slow.

For 330 and 75 the GCF is 15, so the LCM is 330 × 75 ÷ 15 = 1,650. To reduce the risk of huge intermediate values, divide first: 330 ÷ 15 × 75 gives the same 1,650.

With more than two numbers, apply the rule two at a time, because LCM(a, b, c) = LCM(LCM(a, b), c). Continuing the example, LCM(1,650, 450): the GCF of 1,650 and 450 is 150, so the LCM is 1,650 × 450 ÷ 150 = 4,950. Then LCM(4,950, 225): 225 divides 4,950 exactly, so the answer stays 4,950. The results table lists each step.

The LCM appears in everyday scheduling problems. If one bus leaves every 12 minutes and another every 18 minutes, and they leave together now, they next leave together after LCM(12, 18) = 36 minutes. It is also exactly what you need to add fractions with different denominators.

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