LCM

In this section, we learn how to find the LCM or Least Common Multiple for a group of numbers. We will use the LCM in the next section when we begin adding and subtracting fractions. First and foremost we learn about multiples. To obtain the multiples of a number we multiply the number by each non-zero whole number.
Example 1: List the multiples of 3
• Multiply 3 by each non-zero whole number
• 3 x 1 = 3, 3 x 2 = 6, 3 x 3 = 9, 3 x 4 = 12...
Multiples of 3: 3, 6, 9, 12, 15, 18, 21,...
Common Multiples are multiples that are common to all numbers of a group.
Example 2: List the first few common multiples of 3 and 12
•Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42,...
•Multiples of 12: 12, 24, 36, ...
Common Multiples: 12, 24, 36,...
The Least Common Multiple or LCM is the smallest multiple that is common to a group of numbers. We build the LCM from the prime factorization of each number.
LCM - Least Common Multiple

1. Factor each number completely
2. Generate a list that contains each prime factor from all numbers of the group
3. When a prime factor is repeated between two or more numbers, include only the largest number of repeats from any of the prime factorizations
4. The LCM is the product of the numbers on the list

Example 3: Find the LCM of 18, 20, and 36
-Factor each number completely
18 = 2 x 3 x 3
20 = 2 x 2 x 5
36 = 2 x 2 x 3 x 3
-Generate a list that contains each prime factor from all numbers of the group
2, 3, and 5 are the only prime factors involved in all numbers
-Include the largest number of repeats from any of the prime factorizations
2 - Include 2 factors
3 - Include 2 factors
5 - Include 1 factor
-LCM is the product of the numbers on the list:
2 x 2 x 3 x 3 x 5 = 180
LCM(18, 20, 36) = 180