LCM Calculator (Least Common Multiple)
Type two or more whole numbers to find the smallest number they all divide into, with the prime factors and the check shown.
Two to 12 numbers, separated by commas or spaces.
LCM of 4, 6, 10
60
60 is the smallest number that 4, 6, 10 all divide into.
- LCM in primes
- 2² × 3 × 5
How to find the least common multiple
The least common multiple (LCM) is the smallest positive number that each of the numbers divides into exactly. For 4, 6 and 10 it is 60: the multiples of 10 are 10, 20, 30, 40, 50, 60, and 60 is the first one that 4 and 6 also divide.
Listing multiples works for small numbers. The reliable way is prime factors: 4 = 2², 6 = 2 × 3 and 10 = 2 × 5. Take every prime that appears in any of the numbers, at its highest power: 2² × 3 × 5 = 60. Compare this with the GCF, which takes only the shared primes at their lowest power.
For two numbers there is a shortcut through the GCF: LCM(a, b) = a × b ÷ GCF(a, b). For 21 and 6 the GCF is 3, so the LCM is 21 × 6 ÷ 3 = 42. For more numbers, apply it two at a time: LCM(4, 6) = 12, then LCM(12, 10) = 60.
lcm(a, b) = a × b ÷ gcd(a, b)
- US grade 6 standards ask students to find the least common multiple of two whole numbers up to 12, and the greatest common factor of two numbers up to 100 (6.NS.B.4). Source: Common Core State Standards, Grade 6 The Number System (6.NS.B.4).
- For two positive integers, the product of their GCD and LCM equals the product of the numbers. Source: Wikipedia, Least common multiple.
The LCM of 1 to n
The smallest number that every whole number from 1 to n divides; it grows quickly because each new prime power multiplies it.
| n | LCM of 1 to n | In primes |
|---|---|---|
| 2 | 2 | 2 |
| 3 | 6 | 2 × 3 |
| 4 | 12 | 2² × 3 |
| 5 | 60 | 2² × 3 × 5 |
| 6 | 60 | 2² × 3 × 5 |
| 7 | 420 | 2² × 3 × 5 × 7 |
| 8 | 840 | 2³ × 3 × 5 × 7 |
| 9 | 2,520 | 2³ × 3² × 5 × 7 |
| 10 | 2,520 | 2³ × 3² × 5 × 7 |
| 11 | 27,720 | 2³ × 3² × 5 × 7 × 11 |
Show all 20 rowsShow fewer
| 12 | 27,720 | 2³ × 3² × 5 × 7 × 11 |
| 13 | 360,360 | 2³ × 3² × 5 × 7 × 11 × 13 |
| 14 | 360,360 | 2³ × 3² × 5 × 7 × 11 × 13 |
| 15 | 360,360 | 2³ × 3² × 5 × 7 × 11 × 13 |
| 16 | 720,720 | 2⁴ × 3² × 5 × 7 × 11 × 13 |
| 17 | 12,252,240 | 2⁴ × 3² × 5 × 7 × 11 × 13 × 17 |
| 18 | 12,252,240 | 2⁴ × 3² × 5 × 7 × 11 × 13 × 17 |
| 19 | 232,792,560 | 2⁴ × 3² × 5 × 7 × 11 × 13 × 17 × 19 |
| 20 | 232,792,560 | 2⁴ × 3² × 5 × 7 × 11 × 13 × 17 × 19 |
| 21 | 232,792,560 | 2⁴ × 3² × 5 × 7 × 11 × 13 × 17 × 19 |
Frequently Asked Questions
How is the LCM used with fractions?
The least common denominator of two fractions is the LCM of their denominators. To add 3/4 and 1/6, the LCM of 4 and 6 is 12, so rewrite them as 9/12 and 2/12.
When do two numbers have an LCM equal to their product?
When they are coprime (their GCF is 1). The LCM of 8 and 9 is 72; the LCM of 8 and 12 is only 24, because they share a factor of 4.
Two buses leave together, one every 12 minutes and one every 15. When do they next leave together?
After LCM(12, 15) minutes: 12 = 2² × 3 and 15 = 3 × 5, so the LCM is 2² × 3 × 5 = 60 minutes.