Before understanding the greatest common factor, we first need to know what a factor is.
What Is a Factor?
A factor is an integer that appears in a multiplication operation. So, if you ask, "What are the factors of a number?", you are actually asking, "Which integers multiplied together equal this number?"
For example, 4 and 6 multiplied together equal 24. This means that 4 and 6 are both factors of 24. You might say that 2 and 12 multiplied together also equal 24.
That's right, they are both factors of 24. A number can have more than two factors.
Let's list which integers multiply to 24:
1 * 24 = 24,
2 * 12 = 24,
3 * 8 = 24,
4 * 6 = 24.
Thus, we have the table of all factors of 24: [1, 2, 3, 4, 6, 8, 12, 24].
What Are Trivial Factors?
Did you notice that 24 has two special factors: 1 and 24? These are the "trivial factors" of 24. Because any number multiplied by 1 equals itself, this means that 1 and the number itself are always factors of itself.
What Are the Common Factors?
Simply put, a common factor is a factor shared by two or more numbers. For example, let's take the number 12. Now, let's find all the factors of 12:
1 * 12 = 12,
2 * 6 = 12,
3 * 4 = 12.
Therefore, all the factors of 12 are [1, 2, 3, 4, 6, 12]. Comparing this to the factors of 24, we find that the same numbers are [1, 2, 3, 4, 6, 12]. That's right, these are all common factors of 24 and 12.
What Is the GCF?
At this point, everything becomes simple. As the name suggests, it is finding the largest common factor among the numbers. In the table of common factors of 24 and 12, the largest number is 12. 12 is their GCF.
How to Find The GCF?
1. Enumeration of Factors
We used the enumeration method in the previous example. First, we need to list all the factors of the numbers, then find the common factors. Finally, find the largest common factor among the common factors. It's that simple!
2. Euclidean Algorithm
Divide the larger number by the smaller number, then divide the divisor by the remainder, and so on, until the remainder is 0. The last divisor is the greatest common factor (GCF). For example, to find the GCF of 18 and 12: 18 ÷ 12 = 1, remainder 6. Go on: 12 ÷ 6 = 2, remainder 0. Therefore, the GCF of these two numbers is 6.
The Use of The GCF
Why do you need the greatest common factor? One of the most important uses of the GCF is simplifying fractions. A fraction is essentially composed of two numbers. If the numerator and denominator of a fraction have a common factor, it means that the fraction is not in its simplest form. For example, 12/24 can be simplified to 1/2 by dividing both the numerator and denominator by 12. Isn't that much simpler?
Final Thoughts
In summary, the greatest common factor (GCF) can help you simplify fractions in one step. We hope our content helps you understand the concept of the GCF.
