What the LCM Is For
The least common multiple is the smallest number that every value in a set divides into. Its most common use is adding fractions: to add 1/12 and 1/18 you need a common denominator, and the LCM of 12 and 18 is 36 — the smallest one that works.
Three Methods
Listing multiples works for small numbers. Write out the multiples of each and find the first they share. Simple and slow.
Prime factorisation is the general method. Factorise each number, then take the highest power of every prime that appears:
| Number | Prime factorisation |
|---|---|
| 12 | 2² × 3 |
| 18 | 2 × 3² |
| 30 | 2 × 3 × 5 |
| LCM | 2² × 3² × 5 = 180 |
Using the GCF is fastest for two numbers:
LCM(a, b) = a × b / GCF(a, b)
For 12 and 18: 216 / 6 = 36. This identity holds only for two numbers — extending it to three requires applying it pairwise.
Where It Shows Up
- Adding fractions — the least common denominator is the LCM of the denominators.
- Repeating schedules — two buses running every 12 and 18 minutes coincide every 36 minutes.
- Gear ratios — how many turns before two gears return to their starting alignment.
- Cicada life cycles — 13 and 17 year cycles are prime, which maximises the LCM with any predator cycle and minimises how often they coincide.
Useful Properties
- LCM(a, b) is never smaller than the larger of the two numbers.
- If one number divides the other, the LCM is the larger: LCM(4, 12) = 12.
- For coprime numbers, the LCM is simply the product: LCM(8, 9) = 72.
- LCM(a, b) × GCF(a, b) = a × b.
Frequently Asked Questions
What is the LCM of a number and 1?
The number itself, since every integer is a multiple of 1.
Can the LCM be smaller than one of the numbers?
No. It must be a multiple of every input, so it is at least as large as the largest one.
How do I find the LCM of three or more numbers?
Prime factorisation handles any count at once. Alternatively, compute the LCM of the first two, then the LCM of that result with the third, and so on.