How do you find the LCM?

Answer 1

Least Common Multiple of Two Numbers

The "least common multiple" is the smallest integer that contains both numbers as factors. It is the product of the two numbers divided by any common factors.

To determine the least common multiple of two numbers, determine the prime factors of both numbers. Then, determine the prime factors they have in common. Multiply the numbers together, and divide by the prime factors they have in common (the product of these is their "greatest common factor").

Example: Find the least common multiple of 12 and 15.

The prime factors of 12 are 2 , 2, and 3 (12 = 2 x 2 x 3)

The prime factors of 15 are 3 and 5 (15 = 3 x 5)

The only prime factor in common is 3.

The least common multiple is (12 x 15) divided by 3. This is 180 / 3 = 60

The least common multiple is 60.

Example: Find the least common multiple of 9 and 11.

The prime factors of 9 are 3 and 3.

The prime factors of 11 are only 11.

There are no prime factors in common.

The least common multiple is 9 x 11 = 99.

Example: Find the least common multiple of 30 and 42.

The prime factors of 30 are 2, 3, and 5.

The prime factors of 42 are 2, 3, and 7.

The prime factors in common are 2 and 3.

The least common multiple is (2 x 3 x 5) x (2 x 3 x 7) / (2 x 3) = 210. By representing it as a calculation with prime factors, you can cancel out the divisors, so you have as your reduced calculation 5 x 2 x 3 x 7 rather than 30 x 42 ÷ 6.

Least Common Multiple of Three or More Numbers

For one method, check the related question "How do you find the least common multiple of three numbers?" in the links below.

If you are determining the least common multiple for three or more numbers, it is more complicated because you must divide by prime factors that all or even just a pair of the numbers have in common. This paragraph is not a thorough description of the process, but only intends to give an idea of using prime factors to determine the LCM of three or more numbers.

Another method to find the least common multiple of more than two numbers is to take two numbers and determine their least common multiple. Then, take that number and one of the other numbers and determine their least common multiple. Continue calculating the least common multiple, two numbers at a time. If there are four or more numbers, you can find the least common multiple for each pair of numbers, and then the least common multiples of those results.

Example: Find the least common multiple of 4, 7, and 9.

The prime factors of 4 are 2 and 2.

The prime factors of 7 are 7.

There are no prime factors in common, so the least common multiple of 4 and 7 is 4 x 7 = 28. Now, find the least common multiple of 28 and 9.

The prime factors of 28 are 2, 2, and 7.

The prime factors of 9 are 3 and 3.

There are no prime factors in common, so the least common multiple of 28 and 9 is 28 x 9 = 252.

The least common multiple of 4, 7, and 9 is 252.

Example: Find the least common multiple of 2, 3, 7, 8, and 10.

Start with the first pair of numbers, 2 and 3.

The prime factors of 2 are 2.

The prime factors of 3 are 3

There are no prime factors in common, so the least common multiple is 2 x 3 = 6.

Take the next pair of numbers, 7 and 8.

The prime factors of 7 are 7.

The prime factors of 8 are 2, 2, and 2.

There are no prime factors in common, so the least common multiple is 7 x 8 = 56.

Now, find the least common multiple of both results, 6 and 56.

The prime factors of 6 are 2 and 3.

The prime factors of 56 are 2, 2, 2, and 7.

The prime factors in common are a single 2, so the least common multiple is 6 x 56 ÷ 2 = 168.

To finish, find the least common multiple of 168 and the final number, 10.

The prime factors of 10 are 2 and 5.

The prime factors of 168 are 2, 2, 2, 3, and 7.

The prime factors in common are a single 2, so the least common multiple is 168 x 10 ÷ 2 = 840

The least common multiple of 2, 3, 7, 8, and 10 is 840.

Least Common Multiple - Exponential Method

The LCM (least common multiple) is the smallest positive whole number exactly divisible by two or more given whole numbers. Example: the LCM of 14 and 35 is 70 because 70/14=5 and 70/35=2, and no number smaller than 70 is exactly divisible by 14 and 35.

The LCM can also found be for more complex numbers by taking the multiple of the highest power of prime factors from both numbers.

For example, the LCM of 72 and 90 is 360, which is the multiple of the highest power of prime factors from both numbers (23 x 33 x 5 = 2 x 2 x 2 x 3 x 3 x 5 = 360).

Example: LCM 0f 9 and 25 is 225, which is the multiple of the highest power of prime factors in 9 and 25 (32 x 52).

For more than two numbers, LCM can be found by taking the multiple of the highest power of prime factors from all numbers.

For example, the LCM of 28, 70, and 98 is 4,900, which is the multiple of the highest power of prime factors from all three numbers (22 x 52 x 72 = 2 x 2 x 5 x 5 x 7 x 7 = 4900). (Factors: 28 = 22 x 7; 98 = 2 x 72; 350 = 2 x 52 x 7)

Least Common Multiple of One Number

There is no "least common multiple" for a single number, because the least common multiple is the smallest multiple that two or more numbers have in common.

Start by taking the prime factorizations of the numbers. Now, for each unique prime, take the maximum number of times it shows up in your original numbers, and multiply these together.

Example:

LCM of 45, 50, and 16:

45 = 3 * 3 * 5

50 = 2 * 5 * 5

16 = 2 * 2 * 2 * 2

2 shows up at most 4 times

3 shows up at most 2 times

5 shows up at most 2 times

so LCM is 2 * 2 * 2 * 2 * 3 * 3 * 5 * 5 = 3600

Example: 30 and 42

Factor them.

2 x 3 x 5 = 30

2 x 3 x 7 = 42

Select the highest amount of each factor.

2 x 3 x 5 x 7 = 210, the LCM