LCM Calculator

The least common multiple (LCM) of two or more whole numbers is the smallest positive number that every one of them divides into exactly. It is also called the lowest common multiple. The multiples of 4 are 4, 8, 12, 16 and so on, and the multiples of 6 are 6, 12, 18 and so on. The first number in both lists is 12, so the LCM of 4 and 6 is 12. This LCM calculator finds the answer for two or more numbers, shows the prime factor working step by step, and draws a Venn diagram of the prime factors for two numbers.

• The LCM is never smaller than the largest of the numbers.
• If one number is a multiple of the others, the LCM is that number: LCM(3, 6, 12) = 12.
• For coprime numbers the LCM is just their product: LCM(7, 9) = 63.
• GCD(a, b) × LCM(a, b) = a × b, so LCM(a, b) = a × b ÷ GCD(a, b).
• The order of the numbers does not matter.

Write each number as a product of primes, then take every prime that appears and keep its highest power. Multiply those together. Here is 12 and 18.

1. Factorise: 12 = 2² × 3 and 18 = 2 × 3².
2. List the primes that appear: 2 and 3.
3. Take the highest power of each: 2² (from 12) and 3² (from 18).
4. Multiply: 2² × 3² = 4 × 9 = 36.
5. Check: 36 ÷ 12 = 3 and 36 ÷ 18 = 2, and no smaller number works, so LCM(12, 18) = 36.

For two numbers, multiply them and divide by their GCD. For 48 and 18 the GCD is 6, so the LCM is 48 × 18 ÷ 6 = 864 ÷ 6 = 144. Dividing before multiplying (48 ÷ 6 × 18 = 8 × 18 = 144) keeps the numbers smaller.

With prime factors, just include every number. For 8, 12 and 20: 8 = 2³, 12 = 2² × 3, 20 = 2² × 5. The highest powers are 2³, 3 and 5, so the LCM is 8 × 3 × 5 = 120. Check: 120 ÷ 8 = 15, 120 ÷ 12 = 10 and 120 ÷ 20 = 6, all whole numbers.

• Adding and subtracting fractions: the LCM of the denominators is the smallest common denominator. For 1/6 + 1/8, LCM(6, 8) = 24, so 4/24 + 3/24 = 7/24.
• Scheduling: if one bus leaves every 15 minutes and another every 20 minutes, they leave together every LCM(15, 20) = 60 minutes.
• Repeating patterns: gears, lights and cycles with different lengths line up again after the LCM of their lengths.
• Problems about the smallest number that leaves the same remainder or is divisible by several numbers.