Greatest Common Factor
In mathematics, the greatest common factor or greatest common divisor of two or more integers is the largest positive integer that divides each of the integers completely.
In this section, we will learn about factors, common factors, and greatest common factor. Before moving to the greatest common factor, first, we will understand factor and common factor.
Factors: Factors are whole numbers that multiplied together to get another number. A number may have more than two factors. For example, 5×3=15, 1×15=15 where 5, 3, 1 and 15 are the factors of 15. Similarly, the factors of 24 are: 1×24=24, 2×12=24, 3×8=24, 4×6=24. Hence, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
Common Factors: The factor(s) that are common in two or more numbers is called the common factor(s). In other words, common factor(s) are numbers that you can multiply together to produce another number. The numbers should divide exactly into two or more numbers. It is necessary to have at least two numbers to find the common factor(s). For example, we have to find the factor of 12 and 16.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16
We see that 1, 2, and 4 is common in both. So, these are the common factors of the integer 12 and 16.
In the above examples, we have observed that 1 and the number itself appears in both the factors. So, we can conclude that 1 and the number itself are the two factors of ever number.
Greatest Common Factor: It is the highest number that completely divides two or more numbers. It is abbreviated for GCF. It is also known as the Greatest Common Divisor (GCD) and the Highest Common Factor (HCF). It is used to simplify the fractions.
How to Find Greatest Common Factor
Follow the steps given below to find the greatest common factor.
- Write all the factors of each number.
- Select the common factors.
- Select the greatest number, as GCF.
We can also use the following formula:
Note: Use the above formula only for two numbers.
Let’s understand it through examples.
Example 1: Find the GCF of 12 and 8.
Solution:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 8: 1, 2, 4, 8
Common Factors: 1, 2, 4
Greatest Common Factor: 4
Hence, the GCF of 12 and 8 is 4.
Example 2: Find the GCF of 24 and 36.
Solution:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common Factors: 1, 2, 3, 4, 6, 12
Greatest Common Factor: 12
Hence, the GCF of 24 and 36 is 12.
Example 3: Find the GCF of 11, 42 and 65.
Solution:
Factors of 11: 1, 11
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Factors of 65: 1, 5, 13, 65
Common Factor: 1
Greatest Common Factor: 1
Hence, the GCF of 11, 42 and 65 is 1.
Example 4: Find the GCF of 126, 172 and 298.
Solution:
Factors of 126: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
Factors of 172: 1, 2, 4, 43, 86, 172
Factors of 298: 1, 2, 149, 298
Common Factors: 1, 2
Greatest Common Factor: 2
Hence, the GCF of 126, 172 and 298 is 2.
Example 5: Find the GCF of 64 and 112.
Solution:
Factors of 64: 1, 2, 4, 8, 16, 32, 64
Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112
Common Factors: 1, 2, 4, 7, 8, 16
Greatest Common Factor: 16
Using GCF Formula
Hence, the GCF of 64 and 112 is 16.
Example 6: Find the GCF of 33 and 56.
Solution:
Factors of 33: 1, 3, 11, 33
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
Common Factors: 1
Greatest Common Factor: 1
Using GCF Formula
Hence, the GCF of 33 and 56 is 1.