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Least common multiple of any set of numbers, three ways, with the working shown.
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The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
A multiple of a number is what you get by multiplying it by a whole number: the multiples of 4 are 4, 8, 12, 16, and so on forever. A common multiple of two numbers appears in both of their lists. The least common multiple is the smallest number in both lists - the first place the two counting patterns meet.
The LCM of 4 and 6 is 12, because 12 is the first number that is both a multiple of 4 and a multiple of 6. That is genuinely all the definition says, and if you ever lose track of a method you can always fall back to writing out the two lists.
Where it earns its keep is adding fractions. To add 1/4 and 1/6 you need a common denominator, and the least common multiple of 4 and 6 gives you the smallest one available - 12, so 3/12 + 2/12 = 5/12. Any common multiple would work; the least one just leaves you less to simplify afterwards.
Type two or more whole numbers separated by commas or spaces - 12, 18 or 4 6 8. Up to ten at once.
The LCM is at the top. Below it, the same answer is reached three ways: prime factorisation, listing the multiples, and the product-over-GCF shortcut. Follow whichever one your class is using.
Each row is a prime and each column a number. The highlighted cell in each row is the highest exponent, which is the one the LCM takes. Multiplying the right-hand column gives the answer.
If a method ever feels like a trick, look at the multiples listing. The first number that appears in every row is the LCM, straight from the definition.
The pairs that come up most often in homework, with the reason rather than just the result.
| Numbers | LCM | Why |
|---|---|---|
| 4 and 6 | 12 | 2² from 4, 3 from 6 |
| 12 and 18 | 36 | 2² from 12, 3² from 18 |
| 3 and 5 | 15 | No shared prime, so the LCM is just 3 × 5 |
| 6 and 8 | 24 | 2³ from 8, 3 from 6 |
| 10 and 15 | 30 | 2 from 10, 3 from 15, 5 from both |
| 8, 9 and 21 | 504 | 2³, 3² and 7 - one prime from each |
| 1 to 10 | 2520 | 2³ · 3² · 5 · 7 |
12, 1812 = 2² · 3 and 18 = 2 · 3². For each prime take the higher exponent: 2² from the 12 and 3² from the 18, giving 2² · 3² = 4 · 9 = 36. The listing confirms it - the multiples of 12 are 12, 24, 36 and the multiples of 18 are 18, 36, so 36 is the first they share. The shortcut agrees too: 12 × 18 = 216, and 216 ÷ 6 = 36.
3, 53 and 5 are both prime and different, so they share no factor. When that happens the LCM is simply the product: 15. This is why the LCM of two primes, or of any two coprime numbers, is never smaller than multiplying them together.
4, 6The LCM of 4 and 6 is 12, so 12 is the least common denominator. Rewriting gives 3/12 + 2/12 = 5/12, which needs no simplifying. Using 24 instead - also a common multiple - would have given 6/24 + 4/24 = 10/24, correct but with an extra reduction step to do.
8, 9, 218 = 2³, 9 = 3², 21 = 3 · 7. Taking the highest power of each prime across all three: 2³ from the 8, 3² from the 9, and 7 from the 21. That gives 8 · 9 · 7 = 504. Note the 3 in 21 contributes nothing, because the 9 already supplies a higher power of 3.
