LCM Calculator

The identity. |A × B| = GCD × LCM, so the LCM is the absolute product divided by the GCD. For 12 and 18 the GCD is 6 and the LCM is 36. For 4 and 6 the GCD is 2 and the LCM is 12. 36 ÷ 12 is 3 and 36 ÷ 18 is 2. Both are whole, which is the check. The GCD calculator shows the divisor alone. This page shows both.

Twelve and 18 meet at 36. Four and 6 meet at 12. The formula divides the product by the gcd so the product does not have to be formed in a way that overflows a casual mental calculation, though these samples are small. The gcd is printed because it is the other half of the identity. Zero is refused. Negatives become a positive multiple. The least common multiple is a positive integer. If you need a signed answer, you are outside the definition used here. Check by confirming that the LCM divided by each input is a whole number. 36 ÷ 12 is 3 and 36 ÷ 18 is 2. Both are whole. That is the test.

After you read 36, divide it by 12 and by 18. Whole quotients mean the multiple is common. The smaller of the two quotients is not special. Both must be whole. The same check on 12 for the inputs 4 and 6 is 12 ÷ 4 = 3 and 12 ÷ 6 = 2. The printed GCD lets you rebuild the product without a calculator: 6 × 36 is 216, and 12 × 18 is 216. If those disagree, you mis-copied a digit. The page uses exact integer arithmetic for these sizes. A result you cannot divide evenly is a copying error, not a new kind of multiple.

The least common multiple of two whole numbers is the smallest positive integer that both divide. For 12 and 18 that integer is 36. For 4 and 6 that integer is 12. The page computes it as the absolute value of the product divided by the greatest common divisor. It also prints the divisor so you can rebuild the product. 12 times 18 is 216, and 6 times 36 is 216. 4 times 6 is 24, and 2 times 12 is 24. If your rebuild disagrees, copy the numbers again. Then divide the multiple by each input. Both quotients must be whole numbers, and there should be no smaller positive integer with that property. Zero is refused. A negative input is folded into a positive multiple, because the least common multiple is defined as positive here. The greatest common divisor page is the other half of the same identity. Use it when you want the divisor without the multiple. Use this page when you want both. The samples are small so the arithmetic fits on one line of paper.

Keep the two worked pairs on a card: 12 with 18 gives 36, and 4 with 6 gives 12. Divide the printed multiple by each input before you leave the page. Both quotients must be whole numbers. Then rebuild the product from the printed divisor and the multiple, and match it to the product of the inputs. Those two checks are enough. The samples stay small so both checks fit on paper without a second tool.

12 and 18

Inputs: A: 12 | B: 18

Leave A as 12 and B as 18, then press Calculate. The LCM is 36.

Result: A: 12 B: 18 LCM: 36 LCM = |A × B| ÷ GCD. The GCD of these two is 6.

4 and 6

Inputs: A: 4 | B: 6

Set A to 4 and B to 6, then press Calculate. The LCM is 12.

Result: A: 4 B: 6 LCM: 12 LCM = |A × B| ÷ GCD. The GCD of these two is 2.

Why is the LCM of 12 and 18 equal to 36?

The GCD is 6, and |12 × 18| ÷ 6 is 36.

How do I check it?

36 ÷ 12 is 3 and 36 ÷ 18 is 2. Both are whole numbers.

Why is zero refused?

A multiple of zero is a different question. Both inputs must be non-zero.

Can the inputs be negative?

Yes. The least common multiple is reported positive.

Does order matter?

No.

Are the numbers uploaded?

No. The identity is evaluated in the browser.