When students simplify a fraction or split supplies into equal groups, they’re often using the greatest common factor (GCF), even if they don’t call it that. Finding the GCF is pretty quick. The harder part is explaining why their answer is the greatest, and that’s usually how you can tell if a student really understands the concept or just got lucky. There are two ways to find it that make students show their reasoning: listing factors and comparing prime factorizations.
What Is the Greatest Common Factor
For positive whole numbers, the greatest common factor (GCF) is the biggest number that divides into each of them evenly with no remainder. You’ll also see it called the highest common factor or the greatest common divisor. A factor divides into a number, and a multiple is what you get when you multiply that number by a whole number. Students mix these two up a lot, so it’s worth keeping them straight from the start.
Find the GCF by Listing Factors
Say you want the GCF of 18 and 24. Start by writing out every positive factor of each number. Going through factor pairs helps you not miss any. For 18, the pairs are 1 and 18, 2 and 9, and 3 and 6.
Factors of 18: 1, 2, 3, 6, 9, 18.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Now compare the two lists. The factors they share are 1, 2, 3, and 6. The biggest one is 6, so the GCF is 6. Watch for students who stop at 3. It does divide both numbers, but it’s only a common factor, not the greatest one. This method works well when the numbers are small and the factor lists are short.
Find the GCF Using Prime Factorization
Prime factorization means breaking a number down into prime numbers multiplied together. For 24 and 36, you’d write:
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
Next, match up the prime factors that show up in both lines. Both numbers have two 2s and one 3 in common, so multiply those together:
2 × 2 × 3 = 12
So the GCF is 12. The rule is to use the smaller count of each shared prime. You can’t take a third 2 because 36 only has two of them, and you can’t take a second 3 because 24 only has one. The nice part about this method is you don’t have to list out every factor.
Use the GCF to Simplify a Fraction
To simplify a fraction, divide the numerator and denominator by their GCF. For 18 over 24, the GCF is 6:
1824=18 ÷ 624 ÷ 6=34
Dividing the top and bottom by the same nonzero number doesn’t change the fraction’s value. The result is in simplest form because 3 and 4 don’t share any factor bigger than 1. If a student divides by 3 first, they get 6 over 8. That’s equivalent, but they’d still have to simplify it again.
Turn the Calculation Into a Classroom Problem
Ask students to split 24 pencils and 36 erasers into as many identical supply kits as they can, with nothing left over. The number of kits has to divide evenly into both numbers. Since the GCF of 24 and 36 is 12, they can make 12 kits with 2 pencils and 3 erasers in each.
Then have them explain why 6 kits would also work but isn’t the most they can make. To check their answer, they can multiply what’s in one kit by 12 and make sure they get back to 24 pencils and 36 erasers. That way they can see what the 12 actually stands for.
Check the Answer With a GCF Calculator
After students solve a problem, they can use Julius’s free GCF calculator to check their answers and compare steps. It takes two or more numbers and runs right in the browser, no account needed. For example, if you enter 12, 18, and 30, you’ll get 6, because all three numbers have 1, 2, 3, and 6 as common factors. I’d still have students explain which factors are shared, so they’re checking their method and not just the answer.
Common Questions
Can the GCF be 1?
Yes. The numbers 8 and 15 only have 1 as a common factor. Numbers like that are called coprime, even though neither one is actually prime.
Can you find the GCF of three numbers?
Yes. You can compare the factors of all three, or find the GCF of the first two and then find the GCF of that answer and the third number.
