GCF, also known as (Greatest Common Factor, Greatest Common Divisor, Highest Common Factor) of a set of numbers is the largest number that is the factor of all those numbers. In other words, GCF is the highest number that divides exactly into two or more numbers. It is the “highest” thing for simplifying fractions.
Let us understand GCF with an example:
Table of Contents
GCF Example
What is the Greatest Common Factor (GCF) of 12 and 16?
Following are the steps to find the Greatest Common Factor (GCF) of 12 and 16:
Step 1: Find the factors of 12.
Factors of Twelve (12) are: 1, 2, 3, 4, 6, and 12
Step 2: Find the factors of 16
Factors of Sixteen (16) are:1, 2, 4, 8, and 16
Step 3: List the common factors of 12 and 16.
Common factors of 12 and 16 are:1, 2, and 4
Hence, 4 is the GCF or Greatest Common Factor of 12 and 16.
How to Find GCF of 24 and 108 Using Prime Factorization?
Following are the steps to find the Greatest Common Factor (GCF) of 24 and 108 using prime factorization:
Step 1: Find the prime factors of 24
Prime Factors of Twnety-Four (24) are: 2 2 2 3
Step 2. Find the prime factors of 108
Prime Factors of One hundred eight (108) are: 2 2 3 3 3
Step 3. List the common prime factors of 24 and 108
Common Prime factors of 24 and 108 are: 2 2 3 = 12
Hence,12 is the GCF or Greatest Common Factor of 24 and 108
Why is GCF Useful?
One of the most useful things we can do with the GCF is simplifying fractions. Let’s see how can we simplify:
Simplify fraction
In our previous example, we have found that 1,2, and 4 are the common factors of 12 and 16 and so the Greatest Common Factor is 4. Therefore, the largest number that can exactly divide both 12 and 16 is 4, as shown below:
=
As the GCF of 12 and 16 is 4,
Accordingly, can be simplified to .
What is Prime Factorization?
In Mathematics, prime factorization of a number X is the set consisting of prime numbers whose product is equal to the original number i.e. X.
Example: What is the prime factorization of 18?
Prime factorization of 18 is 2 3 3 = 2 3²
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Prime Factorization Definition
Prime factorization is a method of splitting a number into its prime numbers that help us to form the original numebr when multiplied. In other word, we multiply the prime factors to obtain the original number in this method.
Example: What is the prime factorization of 18?
Prime Factors of 18 are 2, 3, and 3
Accordingly,
Prime factorization of 18 is 2 3 3 = 2 3²
Is It Easy to Find Prime Factorization of Larger Numbers?
Yes, it is quite easy to find the prime factorization if larger numbers. It can be done sby starting with small prime numbers. For example, while finding the factors of number 72, we start by dividing 72 by 2, which gives us the result 36. Now, we can see 36 can be easily divided by 2, which gives us the result 18. Again 18 can be divided by 2, which gives us the result 9. Now, the number 9 cannot be divide by 2. Hence, we will look for next prime number that can divide 9 completely. The next prime number which can divide 9 completely is 3, which gives us the result 3.
Hence, the prime factorization of 72 is 2 2 2 3 3 = 2³ 3²