Greatest Common Factor
The largest factor that any given numbers have in common.
Example: Factors of 4 = {1, 2, 4}. Factors of 6 = {1, 2, 3, 6}. The GCF of 4 and 6 is {2}.
Factoring can be used to find greatest common factors and to find least common multiples.
Example 1: Find the GCF Using a List of Common Factors
One way to find the Greatest Common Factor (GCF) is to list all the factors of the given numbers. The largest number that they have in common is the GCF. Look at the example below.
Find the GCF for 24 and 36.
Factors of 24 = {1, 2, 3, 4, 6, 8, 12, 24}
Factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}
The largest number in both lists is 12. Therefore, 12 is the Greatest Common Factor of 24 and 36.
Example 2: Find the GCF Using Factor Trees and Prime Factorization
Another way to find the GCF is to identify the prime factors for each of the numbers. If a prime factor is used more than once, it is multiplied out to its highest value. Look at the example below.
Find the GCF for 35 and 56.
The GCF is 7.
Example 3: Find the GCF Using a List of Prime Factors
Find the GCF for 16 and 24.
First, find the prime factorization for both numbers. You can use either a factor tree or a list, whichever is easiest for you!
16 = 2 × 2 × 2 × 2
24 = 2 × 2 × 2 × 3.
Second, identify the prime factors that both numbers have in common and multiply them. With 16 and 24, the common prime factors are {2 x 2 x 2} which equals 8. The GCF of 16 and 24 is 8.
Example 4: Find GCF of More Than Two Numbers
Find the GCF for 12, 18 and 36.
First, find the prime factorization for all the numbers. You can use either a factor tree or a list ...whichever is easiest for you!
12 = 2 x 2 x 3
18 = 2 x 3 x 3
36 = 2 x 2 x 3 x 3
Second, identify the prime factors that all the numbers have in common and multiply them. With 12, 18, and 36 the common prime factors are {2 x 3}, which equals 6. The GCF of 12, 18, and 36 is 6.
Example 5: Find GCF When There Are NO Common Factors
Find the GCF for 15 and 16.
First, find the prime factorization for all the numbers. You can use either a factor tree or a list, whichever is easiest for you!
15 = 3 x 5
16 = 2 x 2 x 2 x 2
Second, identify the prime factors that all the numbers have in common and multiply them. With 15 and 16 there are NO common factors. This means that the GCF for 15 and 16 is 1. Anytime you are trying to identify GCF between a set of numbers and there are NO common factors, the GCF is always 1.
Reducing Fractions Using GCF
One use of the greatest common factor is when you are reducing fractions to lowest terms. A fraction is in its lowest terms when its numerator and denominator have no common factor other than 1. Sometimes this is called simplifying fractions.
To determine if this fraction is in its lowest terms, the first thing you must do is identify the GCF of both the numerator and the denominator.
7 = (1 x 7)
21 = (3 x 7)
GCF of 7 and 21 = 7
The second step is to divide both the numerator and the denominator by the GCF. The resulting answer will provide you with the fraction in its lowest or simplest terms.
The reason this works is the fraction 7/7 is equal to 1, and any number divided by 1 is equal to itself! Any number over itself is equal to 1, and it is used to reduce fractions to lowest terms.
Let's look at another example!
Simply the following fraction to lowest terms:
Step 1: Find the factors of both 56 and 100. (In this example we will use common factors instead of prime factors.)
56 = (1, 2, 4, 14, 28, 56)
100 = (1, 2, 4, 5, 20, 25, 50, 100)
GCF = 4
Step 2: Divide both the numerator and the denominator by the GCF because 4/4 is the same as dividing by 1. This will not change the value of the fraction. Instead, it will maintain the value of the fraction while simplifying it to lowest terms.
Least Common Multiple
Least Common Multiple - The smallest multiple (other than zero) that any given set of numbers have in common.
Example: Multiples of 4 = {4, 8, 12, 16, ...} Multiples of 6 = {6, 12, 18, 24, ...} The LCM of 4 and 6 is {12}.
This section of the lesson will examine two different ways to identify the Least Common Multiple (LCM) between two or more numbers.
Example 1: Find the LCM Using a List of Multiples.
Find the LCM of 3 and 4.
3 - {3, 6, 9, 12, 15 ...}
4 - {4, 8, 12, 16, ...}
LCM of 3 and 4 = 12
Example 2: Find LCM Using Prime Factorization.
Step 1: Find the prime factorization of each number.
Step 2: Find the common factors.
Step 3: Multiply the common factors and the extra factors.
Example 3: Find LCM of More Than 2 Numbers.
Find the LCM for 12, 20 and 28.
Step 1: Find the prime factorization of each number.
12 = 2 x 2 x 3
20 = 2 × 2 × 5
28 = 2 × 2 × 7
Step 2: Find the common factors.
Step 3: Multiply the common factors and the extra factors.
Notice how using prime factorization to identify the LCM is easier than using a list of common multiples in this example!
The list for each number would have been very long to discover that 420 is the least common multiple between these three numbers.
Example 4: Find LCM of More Than 2 Numbers.
Find the LCM for 6, 9 , 13 and 27.
Step 1: Find the prime factorization of each number.
6 = 2 x 3
9 = 3 x 3
13 = 1 x 13
27 = 3 x 3 x 3
Step 2: Find the common factors.
Step 3: Multiply the common factors and the extra factors.
