Download Greatest Common Factor (GCF)

Version 6/26/2016
Hanh X. Vo
Greatest Common Factor (GCF)
I. Main Use:
Greatest Common Factor (GCF) is used to reduce fractions to their simplest forms
II. Methods for finding GCF:
There are two methods to find GCF.
1) Method 1:
a. List all factors of the numerator of the fraction
b. List all factors of the denominator of the fraction
c. Select the greatest factor that is shared between those two lists. That number is the GCF of
the numerator and denominator
d. Use the found value of GCF to reduce the fraction to its simplest form
Example #1:
Find the simplest form of
24
36
All factors of the numerator (24) are: 1, 2, 3, 4, 6, 8, 12, and 24
All factors of the denominator (36) are: 1, 2, 3, 4, 6, 9, 12, and 36
The greatest factor that is shared between the two lists is 12
Let’s divide both the numerator and denominator by 12:
24 24  12 2


36 36  12 3
Yes, the method works since
2
24
is the simplest form of
3
36
Example #2:
Find the simplest form of
168
252
All factors of the numerator (168) are: 1, 2, 3, 4, 6, 7, 8, 21, 24, 28, 42, 56, 84 and 168
All factors of the denominator (252) are: 1, 2, 3, 4, 6, 7, 9, 12, 18, 28, 36, 42, 63, 72, 84, 108, 126,
and 252
The greatest factor that is shared between the two lists is 84
Let’s divide both the numerator and denominator by 84:
1
Version 6/26/2016
Hanh X. Vo
168 168  84 2


252 252  84 3
Yes, the method works since
168
2
is the simplest form of
3
252
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 in common
c. Multiply each factor the least number of times it occurs in either number
Example #1: Find the GCF of 24 and 36
From the prime factorization we can write:
24 = 2x2x2x3
36 = 2x2x3x3
The two prime factors that are common between the two numbers are: 2 and 3
Factor 2 occurs the least time in number 36 (twice) GCF must have 2x2
Factor 3 occurs the least time in number 24 (once) GCF must have 3
Thus,
GCF=2x2x3=12 ; which is the same as found from Method 1.
Example #2: Find the GCF of 168 and 252
From the prime factorization we can write:
168 = 2x2x2x3x7
252 = 2x2x3x3x7
The three prime factors that are common between the two numbers are: 2, 3 and 7
Factor 2 occurs the least time in number 252 (twice) GCF must have 2x2
Factor 3 occurs the least time in number 168 (once) GCF must have 3
Factor 5 occurs once in both numbers  GCF must have 7
Thus,
GCF=2x2x3x7=84 ; which is the same as found from Method 1.
2