The acronym GCF stands for “greatest common factor.” The greatest common factor (GCF) can be defined as the greatest whole number that has the ability to divide a given number into two or more equal parts. Other names given to GCF are the Greatest Common Divisor (GCD) and the Highest Common Factor (HCF).
The GCF can be a factor of two or more whole numbers. Factors can be defined as digits that can be multiplied among themselves to yield the number itself. A factor that is shared by two or more numbers is referred to as a common factor.
In this article, we will learn more about the GCF of numbers and how to calculate the GCF.
Properties of GCF
• If we have two numbers A and B, then every common divisor of A and B also divides the greatest common divisor (GCD/GCF).
• A divides B if and only if A is the greatest common factor of the pair.
• The GCF of two or more numbers must not be greater than any of the numbers.
• If we have two numbers where one of them is a prime number, then the GCF is either 1 or the prime number itself. For example, the GCF of 4 and 3 is 1. Also, the GCF of 5 and 30 is 5.
• The GCF of two consecutive numbers is 1. For example, the GCF of 5 and 6 is 1.
• The product of the GCF and LCM of any two given numbers is equal to the product of those two numbers.
Methods to Calculate GCF
1. Listing Method
In this method, the first step is to make a note of all the factors of the given numbers. We then proceed to find the common factors of the given numbers. The factor which has the greatest value out of all the common factors becomes the GCF of those numbers. To get a better understanding, let us take a look at an example.
Using the listing method find the GCF of 12 and 16.
• Factors of 12 = 1, 2, 3, 4, 6, 12
• Factors of 16 = 1, 2, 4, 8, 16
• Common Factors = 1, 2, 4
4 holds the greatest value among all the common factors; hence, 4 is the GCF of 12 and 16.
2. Prime Factorization
One of the main disadvantages of using the listing method is that it can prove to be a very time-consuming process when the numbers that are involved are very large. If you have to find the GCF of two numbers, such as 16843 and 3980, using the listing method is a very inefficient way to find the GCF of these two numbers.
Hence, in such a case, we turn to the prime factorization method. The first step is to make a list of all the prime factors of the given numbers. Next, we find the common prime factors. After multiplying the common prime factors, the number that we get is the GCF of the given numbers. Let us see an example given below.
Using the Prime Factorization method find the GCF of 12 and 16.
• Prime Factors of 12 = 2 × 2 × 3
• Prime Factors of 16 = 2 × 2 × 2 × 2
• Common Prime Factors = 2 × 2
Thus, the resultant of 2 × 2 is 4. Hence, the GCF of 12 and 16 is 4.
The concept of GCF is used in several mathematical topics, such as algebra. Hence, it is essential for students to get a good understanding of the chapter. By availing an institution such as Cuemath, a student is sure to excel in his studies and master the topic in no time.