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# Least common multiple

The least common multiple (LCM) is the smallest number that two or more numbers can divide into evenly. To find the LCM, you can use the prime factorization method or list the multiples of each number. Prime factorization involves breaking down numbers into their prime factors and constructing the smallest number with all the factors. Listing multiples involves finding the smallest shared multiple. Both methods help find the LCM of numbers like 18 and 12. Created by Sal Khan.

## Want to join the conversation?

• Is the LCM (Least Common Multiple) useful in real life? If so, could someone provide some examples?
• I'm assuming that if you're baking something, like a really extravagant cake or something, then figuring out the least common multiple just might work when you're trying to figure out how many cartons of eggs to get to satisfy the ingredients. Also, it might work when trying to figure out if a discount is worth it when you're at the supermarket and comparing any of the "2 for \$2" or "5 for \$5" types of deals. It's almost at night in this part of the world right now and I'm a little tired so I could be wrong, but those are my best guesses! :)
• is the least common factor the same as the lcm?
• No. LCM stands for Least Common Multiple. A multiple is a number you get when you multiply a number by a whole number (greater than 0). A factor is one of the numbers that multiplies by a whole number to get that number.

example: the multiples of 8 are 8, 16, 24, 32, 40, 48, 56...
the factors of 8 are 1, 2, 4, 8.

The term least common factor doesn't really make sense since the least common factor of any pair of numbers is 1. Not exactly a useful piece of knowledge.
• finding the LCM is too hard for me! Can anyone give me advice for remembering?
• All you have to do is list the multiplies of both of the numbers and look for the common number.

Example:
5 and 6
5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
The LMC of 5 and 6 is 30.

Example:
10 and 12
10 = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
The LMC of 10 and 12 is 60.

Example:
3 and 7
3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70
The LMC of 3 and 7 is 21.
• what if the number you have has no multiples?
• There is no number without multiples and factors. For factors, there will always be 1 and itself.
eg: factor of a number d = 1 and d
If the number is 1, then the factor is 1.

For multiples, take any number and multiply it with your number.
eg: multiple of a number y = y * another number x

Hope this helps!
• Can we use other methods to find LCM?
• Good question!

There is a prime factorization method for finding the LCM of a list of two or more numbers.

Prime-factor each number. Then for each prime factor, use the greatest number of times it appears in any prime factorization.

Example: Find the LCM of 40, 48, and 72.
40 = 2*2*2*5
48 = 2*2*2*2*3
72 = 2*2*2*3*3

The prime factor 2 occurs a maximum of four times, the prime factor 3 occurs a maximum of two times, and the prime factor 5 occurs a maximum of one time. No other prime factors appear at all.

So the LCM is 2*2*2*2*3*3*5 = 720.

By the way, there is a similar method of finding GCF (or HCF or GCD or HCD, where G means greatest, H means highest, F means factor, and D means divisor), but we use each prime factor the least number of times it appears in any prime factorization. In our example, the GCF would be 2*2*2 = 8.

An interesting property of GCF and LCM is that, for two numbers, the product of the numbers always equals the GCF times the LCM. However, this might not be true for three or more numbers.
• Can the Least common multiple also be the greatest common multiple
• Well yes, in a way. For 0 the only multiple is 0, and nothing else. There's been confusion about 1 having no multiples. But, all the whole numbers are multiples of 1.
• what is the difference between multiple and factor?
• They both involve multiplication. Factors are what you multiply to get a number. Multiples are what we get after multiplying the number by an integer (number, not a fraction). Example: 2 x 2 are factors and 4 is the multiple.
• what is the difference between a multiple and a factor?
• Multiples and factors are both about multiplying, as follows..
A multiple is what you get when you multiply a number by other numbers, like, we get 10, 15, 20, 25, 30 and on and on, which are some MULTIPLES of 5, for example.
(A multiple, is bigger than the number.)
A FACTOR, is what you can multiply together to get a number. Like the factors of 24, can be 1, 2, 3, 4, 6, 8, and 12. These are FACTORS of 24, for example, because you can multiply 2 times 12 to get 24. (The factors of a certain number are smaller than the number.