What the least common multiple means
A multiple of a number is that number times 1, 2, 3 and so on. The least common multiple of a set is the smallest positive number that shows up in every one of their multiple lists. Put another way, it's the smallest number each of them divides into with nothing left over.
Keep both halves of that in mind. The answer has to be common to every number, and it has to be the smallest such number. Most mistakes come from getting one half right and missing the other. Multiplying the numbers together always gives a common multiple, for example, but often not the least one.
Method 1: list the multiples
Write out multiples of each number until one 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. They first meet at 12, so the LCM is 12.
A quick shortcut: only list multiples of the biggest number and test each one against the others. For 9 and 12, try 12 (not divisible by 9), 24 (no), 36 (yes). The LCM is 36. This method is perfect for small numbers and terrible for something like 84 and 90.
Method 2: prime factorization
Break every number into primes. Then, for each prime that appears anywhere, keep the highest power it reaches in any one number. Multiply those powers and you have the LCM.
Take 8, 12 and 15. That's 2^3, 2^2 x 3 and 3 x 5. The highest power of 2 is 2^3, the highest power of 3 is 3, and 5 appears once. So the LCM is 8 x 3 x 5 = 120. This is the method to reach for when you have three or more numbers, because it handles them all at once.
Formula notes
LCM = product of every prime at its highest power in any of the numbers
Method 3: use the GCF
If you already know the greatest common factor, two numbers have a neat shortcut: divide one of them by the GCF, then multiply by the other. For 18 and 24, the GCF is 6, so 18 / 6 x 24 = 3 x 24 = 72.
This works because the GCF is exactly the part the two numbers share, and multiplying them outright would count that shared part twice. Dividing first also keeps the arithmetic small. For a third number, take your answer and repeat the step with it.
Formula notes
LCM(a, b) = a / GCF(a, b) x bLCM(a, b, c) = LCM(LCM(a, b), c)
Which method should you use?
Use listing when both numbers are under about 20 and you can see the answer in a few steps. Use the GCF shortcut for two larger numbers, especially when their common factor jumps out at you, like 10 for 40 and 70. Use prime factorization for three or more numbers, or when a teacher wants to see the work.
All three should agree. If they don't, something went wrong, and that's a useful check on homework. Our LCM calculator shows the prime powers and the step-by-step GCF path side by side, so you can compare your work against both.
Mistakes that give the wrong LCM
The most common one is mixing up LCM and GCF rules. For the LCM you keep each prime at its highest power. For the GCF you keep only shared primes at their lowest power. Swap them and you'll get the GCF by accident.
Other slips are easy to catch with a quick division check. Divide your answer by each original number. If any result isn't a whole number, you dropped a factor. If every result is whole but the results all share a factor (say they're all even), your answer is too big and you can divide it down.
- Using the lowest power of a prime instead of the highest
- Forgetting a prime that appears in only one number
- Multiplying everything together when the numbers share factors
- Not checking that the answer divides evenly by every number
Frequently asked questions
What is the LCM of 8, 12 and 15?
It's 120. In primes those numbers are 2^3, 2^2 x 3 and 3 x 5, so you keep 2^3, 3 and 5 and multiply: 8 x 3 x 5 = 120.
Why do you use the highest power of each prime?
Because the LCM has to be divisible by every number. If one number needs 2^3, any lower power of 2 would leave it short.
What's the fastest way to find the LCM of two numbers?
Find their GCF, divide one number by it and multiply by the other. For 18 and 24 that's 18 / 6 x 24 = 72.
How do I know my LCM isn't too big?
Divide it by each original number. If all of those results share a common factor, divide your answer by that factor and check again.
Do I need the LCM to add fractions?
No, any common denominator works. The LCM just keeps the numbers smallest, so you have less simplifying to do at the end.
References
These sources support the method or guidance used for LCM Calculator. Verify time-sensitive rules at the source.
Try the calculator
Open LCM Calculator, enter your scenario, and compare its supporting rows with this guide's method and checks.