Download 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:
Find the LCD of the two following fractions
1
5
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.
First fraction becomes:
1 1 3 3


8 8  3 24
Second fraction becomes:
5
5  2 10


12 12  2 24
Example #2:
Find the LCD of the two following fractions
1
3
and
12
80
The two denominators are 12 and 80
1
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.
First fraction becomes:
1
1 20
20


12 12  20 240
Second fraction becomes:
3
3 3
9


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.
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