Least Common Multiple (LCM)

Version 6/26/2016 Hanh X. Vo Least Common Multiple (LCM)

I. Main Use: Least Common Multiple (LCM) is used to find the smallest multiple of two or more numbers. In fractions, the LCM is used to find the Least Common Denominators (LCD) when adding or subtracting fractions with different denominators.

II. Methods for finding LCM:

There are two methods to find LCM.

1) Method 1:

a. List some multiples of the denominator of the first fraction b. List all factors of the denominator of the second fraction c. Select the smallest multiple that is shared between those two lists. That number is the LCM of the two denominators. The LCM value is also the Least Common Denominator of the two fractions

Example #1: 1 5 Find the LCD of the two following fractions and 8 12 The two denominators are 8 and 12 Some multiples of the denominator of the first fraction (8) are: 8, 16, 24, 32, 40, 48, … Some multiples of the denominator of the second fraction (12) are: 12, 24, 36, 48, … The smallest multiple that is shared between the two lists is 24. This is the LCM of the two numbers 8 and 12. The value 24 is also the Least Common Denominator (LCD) of the two given fractions.

1 13 3 First fraction becomes:   8 83 24

5 5 2 10 Second fraction becomes:   12 12 2 24

Example #2: 1 3 Find the LCD of the two following fractions and 12 80 The two denominators are 12 and 80

Version 6/26/2016 Hanh X. Vo

Some multiples of the denominator of the first fraction (12) are: 12, 24, 36, 48, 60, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264 … Some multiples of the denominator of the second fraction (80) are: 80, 160, 240, 320, … The smallest multiple that is shared between the two lists is 240. This is the LCM of the two numbers 12 and 80. The value 240 is also the Least Common Denominator (LCD) of the two given fractions.

1 1 20 20 First fraction becomes:   12 12 20 240

3 33 9 Second fraction becomes:   80 803 240

2) Method 2: Using Prime Factorization

a. Use Prime Factorization to find all the prime factors for the two numbers b. List all factors that are present in either number c. Multiply each factor the greatest number of times it occurs in either number

Example #1: Find the LCM of 8 and 12 From the prime factorization we can write: 8 = 2x2x2 12 = 2x2x3 The two prime factors that are present in the two numbers are: 2 and 3 Factor 2 occurs the greatest number of times in number 8 (three times) LCM must have 2x2x2 Factor 3 occurs the greatest number of times in number 12 (once) LCM must have 3 Thus, LCM=2x2x2x3=24 ; which is the same as found from Method 1.

Example #2: Find the LCM of 12 and 80 From the prime factorization we can write: 12 = 2x2x3 80 = 2x2x2x2x5 The three prime factors that are present in the two numbers are: 2, 3 and 5 Factor 2 occurs the greatest number of times in number 80 (four times) LCM must have 2x2x2x2 Factor 3 occurs the greatest number of times in number 12 (once) LCM must have 3 Factor 5 occurs the greatest number of times in number 80 (once) LCM must have 5 Thus, LCM=2x2x2x2x3x5=240; which is the same as found from Method 1.