Least Common Multiple Calculator

What is LCM?

The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of the numbers without any remainder. In other words, it's the smallest number that all of the numbers can go into. The concept of LCM is particularly useful in solving problems that deal with finding equivalent fractions, adding, subtracting, or comparing fractions with different denominators, and in various problems in arithmetic and algebra where common multiples are needed.

Listing Multiples Method

You list the multiples of the numbers until you get to the first one they have in common.

For example, if you want to find the LCM of 12 and 15:

1. List the multiples of each number:
   • Multiples of 12: 12, 24, 36, 48, 60, 72, ...
   • Multiples of 15: 15, 30, 45, 60, 75, ...
2. Find the first common multiple in the lists:
   • The first multiple that appears in both lists is 60.
   • Thus, the LCM of 12 and 15 is 60.

Prime Factorization Method

Decompose each number into prime factors. The LCM is then the product of the highest power of all prime numbers that appear in any of the factorizations.

Example, for 12 and 18:

• 12 = 2^2 * 3^1
• 18 = 2^1 * 3^2
• LCM = 2^max(2,1) * 3^max(1,2) = 2^2 * 3^2 = 36

Greatest Common Divisor Method

Using the relationship between LCM and GCD of two numbers, where the product of the numbers is equal to the product of their LCM and GCD.

Example, for 21 and 6:

• GCD(21, 6) = 3
• LCM(21, 6) = (21 * 6) / GCD(21, 6) = (126) / 3 = 42