Greatest Common Factor (GCF) Formula, Definition and Example

Have you ever tried to divide something into equal parts and got stuck? Have you ever asked yourself how can I find the biggest number that divides two numbers without leaving any piece behind? That is where the Greatest Common Factor or GCF comes in.

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Greatest Common Factor means the largest number that can divide two or more numbers without leaving a remainder. Some people call it Highest Common Factor or HCF. Others call it Greatest Common Divisor or GCD. Do these names confuse you? Do not worry. All of them mean the same thing.

For example the GCF of 18 and 24 is 6. Why? Because 6 is the biggest number that divides both 18 and 24 equally.

You may ask how do we find GCF in math? There are many ways. Let me show you.

• Listing factors method: Write all factors of both numbers. Choose the biggest one that is common.
• Prime factorization method: Break both numbers into prime numbers and then take the common ones.
• Division method: Keep dividing the bigger number by the smaller one till remainder is zero.
• Euclidean algorithm: A faster division method used in higher classes.
• Venn diagram method: Draw circles of factors and see the common part.

Which method do you like more?

Step by Step Examples

Let us try an example. Find GCF of 18 and 24. Factors of 18 are 1 2 3 6 9 18. Factors of 24 are 1 2 3 4 6 8 12 24. The biggest common number is 6. So the GCF is 6. Easy right?

Now let us try three numbers. Find GCF of 12 16 and 20. Factors of 12 are 1 2 3 4 6 12. Factors of 16 are 1 2 4 8 16. Factors of 20 are 1 2 4 5 10 20. The biggest common number is 4. So the GCF is 4.

Do you see how simple it looks when we break it down?

Why Do We Need GCF?

You may think why should I even learn this? Let me tell you. GCF is used in many real life problems.

• We use GCF to simplify fractions. Example 12/16 becomes 3/4 after dividing top and bottom with 4.
• We use GCF to divide things into smaller equal parts. Like cutting wood or paper into pieces.
• We use GCF to solve ratio problems.
• Even in computers GCF is used in algorithms.

So GCF is not just a math concept. It is also a life skill.

GCF in Algebra

When you move to higher classes GCF is also used in algebra. You will see GCF of monomials. You will see GCF of polynomials. You will learn how to factor expressions using GCF. Without GCF you cannot do algebra easily.

Practice Questions

Want to check yourself? Try these.

1. Find GCF of 45 and 60.
2. Find GCF of 72 and 108.
3. Find GCF of 15 25 and 35.
4. Simplify the fraction 36/48 using GCF.

Can you solve them without looking at the answer key?

GCF Tools

If you get tired of long numbers you can also use online GCF calculators. They show step by step solutions. Some calculators even work with polynomials.

GCF vs LCM

Do not mix GCF with LCM. GCF means the biggest common divisor. LCM means the smallest common multiple. One finds the largest number that divides. The other finds the smallest number that multiplies. Isn’t it amazing how both ideas are opposite yet connected?

Now you know what GCF is. You know how to find it. You know why it is useful. You know how it appears in real life and in algebra. Next time you see numbers ask yourself what is their GCF. Do you see the magic? Numbers have hidden patterns and GCF is the key to unlock them.