The greatest common factor in math is an important concept that students get familiar with at the school level. The greatest common factor (GCD), also known as the highest common divisor (HCD), the greatest common divisor (GCD), or the highest common factor (HCF), has many applications. Sometimes, students encounter fractions that need to be reduced to their lowest terms. In algebra, the knowledge of GCF is required to factorize complex polynomials. Some real-life situations also require us to simplify the ratios of a group of numbers using this concept. Therefore, it is important to understand the concept and properties of the GCF. Unfortunately, students face difficulty in visualizing the concept and associating it with the real world. The reason may be a certain learning disability, a rote-learning approach, or simply a lack of good math teachers.
In this article, we are going to define the greatest common factor and explain different methods to find it with the help of suitable example problems.
Definition of the Greatest Common Factor
According to the Cambridge Dictionary, GCF stands for “greatest common factor”. It is the largest number that can be used to divide a group of numbers exactly (with no remainder).
The following Venn diagram will help you visualize the greatest common factor of two numbers (for example, the GCF of 15 and 18).
The Formula for the Greatest Common Factor
There is no single-step formula to find the GCF. Finding the GCF involves several steps. To understand the concept of the GCF, you need to master factors and multiples first. Therefore, we encourage you to read our blog about factors and multiples and then come back to this article.
How to Find the Greatest Common Factor of a Number – Three Methods with Example Problems
In this section, we will show you how to find the greatest common factor using the list method, the prime factorization (factor tree) method, and the continuous division method.
1. GCF by Listing out the Factors (List Method)
In this method, we write down all the factors/divisors of a group of numbers. After listing down the divisors, we pick the greatest number that commonly divides the said numbers without leaving any remainder. That number is called the highest common factor or the greatest common divisor.
Example 1: Find the greatest common factor of 24 and 40 using the factorization method.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40.
Common factors of 24 and 40: 1, 2, 4, 8.
Therefore, the greatest common factor of 24 and 40 is 8.
2. GCF by Prime Factorization Method (Factor Tree Method)
In this method, we perform the prime factorization for each number and draw a factor tree. We then compare the factor trees of the given numbers and identify their common prime factors. The product of these common prime factors is called the number’s greatest common factor.
A factor tree is a way of expressing the prime factors of a number. Each branch in the tree is divided into factors. No matter how you construct the factor tree, the numbers you end up with at the end of the branches are always the prime factors of the number that you begin with.
Example 2: Find the GCF of 40 and 64 using prime factorization.
Let’s make a factor tree for each number. First, divide the given number by the lowest prime number. Then keep working down and divide any remaining composite factors exactly, until there are no composite factors left.
Now, identify the common prime factors of 40 and 64.
Prime factors of 40 = 2 x 2 x 2 x 5
Prime factors of 64 = 2 x 2 x 2 x 2 x 2 x 2
Now, multiply the common prime factors.
2 x 2 x 2 = 8
Therefore, the GCF of 40 and 64 is 8.
3. GCF by Continuous Division Method
In this method, we divide the larger number by the smaller number using long division. If the remainder is 0, then the divisor is called the GCF. If the remainder is not 0, then we make the remainder of the previous step as the divisor and the divisor of the previous step as the dividend and perform long division repeatedly until we get 0 as the remainder. The divisor of the last division is the GCF of the given two numbers.
Example 3: Find the GCF of 16 and 60 using the continuous division method.
In this example, the divisor of the last division is 4. Therefore, the GCF of 16 and 60 is 4.
The methods mentioned above are not the only methods for finding the GCF of a group of numbers. There are some other tools such as Euclid’s Algorithm and ladder method widely used for this purpose. Research in mathematics education shows that to solve math problems accurately and efficiently, students need to learn multiple strategies as well as how to choose among them. If your child has a strong command of math fundamentals, they can find out the GCF by using any of the above-mentioned methods.
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