Enter two or more positive integers, separated by commas, to find their least common multiple (LCM) — also called the lowest common multiple — with GCD and prime-factorization steps.
When to use LCM instead of GCF
Use LCM when separate repeating patterns need to meet again, or when you need the first total that every group size can divide evenly. Use GCF when you are dividing quantities into the largest possible equal groups without a remainder.
| Clue in the problem | Operation | What the answer represents |
|---|---|---|
| “When will both happen together again?” | LCM | First shared time or cycle count |
| “What is the smallest total both package sizes fit?” | LCM | First shared multiple |
| “What is the largest equal group size?” | GCF | Largest shared factor |
| “How many identical groups can I make?” | Often GCF | Greatest number or size of equal groups, depending on the wording |
Worked schedule example: two repeating buses
Two buses leave the same stop together at 7:30 AM. One returns every 12 minutes and the other every 18 minutes. When will they next leave together?
Multiples of 12: 12, 24, 36, 48, ...
Multiples of 18: 18, 36, 54, ...
LCM(12, 18) = 36
7:30 AM + 36 minutes = 8:06 AM
Enter 12, 18 in the calculator. The GCD is 6, so the two-number check is
(12 × 18) ÷ 6 = 36. Both schedules are measured from the same 7:30 AM
starting event; if their starting times differ, finding an LCM alone does not
give the next clock time.
Compare with a GCF grouping problem
Suppose 24 red counters and 36 blue counters must be split into the greatest number of identical groups, using every counter. This asks for GCF(24, 36) = 12, so there are 12 groups with 2 red and 3 blue counters in each. LCM would answer a different question: the smallest positive total that is a multiple of both 24 and 36, which is 72.
The Common Core 6.NS.B.4 standard includes finding greatest common factors and least common multiples of whole numbers. Illustrative Mathematics' florist-order activity uses two package sizes and lists their multiples to identify the first shared order quantity. Those sources establish the instructional context; the bus and counter examples above are original CalculatorBuddy examples and have not been classroom tested.
Quick check prompts
- Two lights flash together, then every 8 seconds and 14 seconds. How many seconds pass before they flash together again? Why is this an LCM problem?
- A teacher has 28 pencils and 42 erasers. What is the greatest number of identical supply sets that can use everything? Why is this a GCF problem?
- For the bus example, explain why 18 is not the answer even though it is one bus's interval.
Answer check: LCM(8, 14) = 56 seconds. GCF(28, 42) = 14 sets, each with 2 pencils and 3 erasers. Eighteen is not divisible by 12, so only one bus has completed a whole cycle after 18 minutes.
Examples
Default list 330, 75, 450, 225
Prime factors: 330 = 2 × 3 × 5 × 11, 75 = 3 × 5 × 5, 450 = 2 × 3 × 3 × 5 × 5, and 225 = 3 × 3 × 5 × 5. Highest powers are 2, 3², 5², and 11, so the LCM is 4950. The GCD of the same list is 15.
Page example 18, 26
GCD(18, 26) = 2, so LCM = (18 × 26) / 2 = 234. Prime factors 2 × 3 × 3 × 13 give the same result.
Page example 21, 14, 38
LCM(14, 38) = 266, then LCM(266, 21) = 798. The primes 2, 3, 7, and 19 multiply to the same value.
Formula
The LCM is the smallest positive integer divisible by every listed number.
Prime factorization. Factor each integer, keep the highest power of each prime, and multiply:
LCM(21, 14, 38)
21 = 3 × 7
14 = 2 × 7
38 = 2 × 19
LCM = 2 × 3 × 7 × 19 = 798
Greatest common divisor. For two positive integers a and b,
LCM(a, b) = (a × b) / GCD(a, b)
Extra integers fold in the same way: LCM(a, b, c) = LCM(LCM(a, b), c).
LCM and GCF are inverses for two numbers: LCM(a, b) × GCF(a, b) = a × b. Use the GCF calculator when you want the greatest common factor.