Least Common Multiple Calculator

The least common multiple (LCM) of two numbers is the smallest number that both given numbers can divide evenly into. For example, the LCM of 12 and 24 is 24, since both 12 and 24 will divide evenly into 24 and no smaller number. The LCM of 60 and 61 is 3660, which is the product of 60 and 61. You can use the calculator on the left to compute the LCM, or follow the explanation below.

How to Compute the LCM by Hand

Example 1: Find the greatest common factor of 30 and 32. The easiest way to do this is to find the prime factorization of both numbers using a factor tree, then calculate the product of all the common prime factors in the union of the two sets. The prime factorizations of 30 and 32 are

30 = 2*3*5

32 = 2*2*2*2*2

The union of the sets {2, 3, 5} and {2, 2, 2, 2, 2} is {2, 2, 2, 2, 2, 3, 5}. Notice that 2 appears in this set five times since the maximum multiplicity of this factor is five, in the number 32. In other words, 32 has the highest number of 2's (five of them). The factors 3 and 5 each appear once since they occur no more than once in either number.

Thus, the least common multiple of 30 and 32 is 2*2*2*2*2*3*5 = 480.


Example 2: Of the three numbers 100, 66, and 45, which pair has the smallest LCM? For this problem we need to compute LCM(100, 66), LCM(100, 45), and LCM(66, 45). To start, let's find the prime factorization of each number:

100 = 2*2*5*5

66 = 2*3*11

45 = 3*3*5

Thus,

LCM(100, 66) = 2*2*3*5*5*11 = 3300

LCM(100, 45) = 2*2*3*3*5*5 = 900

LCM(66, 45) = 2*3*3*5*11 = 990

So the pair of 100 and 45 has the smallest LCM.

Relation Between GCD and LCM

The product of the LCM and GCD of two numbers is equal to the product of the two numbers. That is,

LCM(a,b)*GCD(a,b) = ab

The main consequence of this expression is that you can always find the GCD if you know the LCM and vice versa:

LCM(a,b) = ab/GCD(a,b)

GCD(a,b) = ab/LCM(a,b)

For example, since the GCD of 30 and 32 is 2, you can compute the LCM of the two numbers more quickly:

LCM(30, 32) = 30*32/2 = 480

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