How do you find the GCF of numbers?
Write each number as a product of primes, keep only the primes that appear in every number, and use the lowest power of each. Multiply those together to get the GCF.
Here is how to find GCF for 24, 36 and 60 using prime factorization.
- Factor each number: 24 = 2³ × 3, 36 = 2² × 3², 60 = 2² × 3 × 5.
- Find the primes all three share: 2 and 3. The 5 only appears in 60, so it's left out.
- Take the lowest power of each shared prime: 2² and 3.
- Multiply: 4 × 3 = 12.
Check it: 24 ÷ 12 = 2, 36 ÷ 12 = 3 and 60 ÷ 12 = 5. The quotients 2, 3 and 5 share no factor, which tells you nothing bigger than 12 would work.
Can you find the greatest common factor by listing factors?
Yes. List every factor of each number, circle the ones on every list and pick the largest. This is the method most classes teach first.
For 18 and 24, the factors of 18 are 1, 2, 3, 6, 9, 18 and the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The common factors are 1, 2, 3 and 6, so the greatest common factor is 6.
Listing works well for numbers under 50 or so. For larger numbers it's easy to miss a factor, and that's where prime factorization or the Euclidean algorithm is safer.
How does the Euclidean algorithm find the GCD?
Divide the larger number by the smaller one, then replace the larger number with the remainder. Repeat until the remainder is 0. The last nonzero remainder is the GCD.
This is the method a GCD calculator uses for big numbers, because it never needs you to factor anything. Here it is for 48 and 18.
For three or more numbers, find the GCF of the first two, then the GCF of that answer and the next number. For 24, 36 and 60: 36 = 1 × 24 + 12, then 24 = 2 × 12 + 0, so the GCF of 24 and 36 is 12. Since 60 = 5 × 12 + 0, the GCF of all three stays 12.
What does it mean when the GCF is 1?
A GCF of 1 means the numbers share no factor except 1. Numbers like this are called relatively prime, or coprime.
For example, 17 and 31 are both prime, so their GCF is 1. The numbers don't have to be prime themselves, though. 8 and 9 are each composite, and their GCF is still 1 because 8 = 2³ and 9 = 3² share no prime.
When you see a GCF of 1 in a fraction problem, the fraction is already in lowest terms.
How do you use the GCF to simplify fractions?
Divide the numerator and the denominator by their GCF. The result is the fraction in lowest terms, reached in a single step.
To simplify 84/126, find the GCF of 84 and 126, which is 42. Then 84 ÷ 42 = 2 and 126 ÷ 42 = 3, so 84/126 = 2/3. Dividing by a smaller common factor like 2 or 6 also works, but you'd have to repeat it several times.
How is the GCF used to factor expressions?
Factoring out the GCF is the first step in factoring almost any polynomial. You find the GCF of the coefficients (and any shared variable), write it outside parentheses and divide each term by it.
For 12x + 18, the GCF of 12 and 18 is 6. Divide each term by 6 to get 2x and 3, so 12x + 18 = 6(2x + 3). Multiply back out to check. The factor expression calculator handles longer polynomials if you need more than the number part.
What are common mistakes when finding the GCF?
The biggest one is using the highest power of each prime instead of the lowest. That gives you the LCM, not the GCF. For 24 and 36, taking 2³ × 3² gives 72, which can't be right because the GCF can never be larger than the smallest number.
Another mistake is keeping a prime that only some of the numbers have. In 24, 36 and 60, the prime 5 belongs to 60 alone, so it stays out of the GCF.
Students also stop at the first common factor they find. Seeing that 2 divides 24 and 36 doesn't mean 2 is the greatest common factor. Always check whether the quotients still share a factor.
How do you use this greatest common factor calculator?
Type 2 to 8 whole numbers separated by commas, such as 24, 36, 60, and press solve. The answer appears first, followed by the working.
The steps come in a fixed order. First you see the prime factorization of each number. Next comes the list of shared primes at their lowest powers, then the Euclidean algorithm as a second check. The last step divides each number by the GCF so you can see that every quotient is whole.
If you're practicing for a test, try the problem on paper first and compare your steps line by line. The prime factor step is the one most worth copying, since the same factorizations help with the LCM.
| Numbers | GCF | LCM |
|---|---|---|
| 12, 18 | 6 | 36 |
| 48, 18 | 6 | 144 |
| 24, 36, 60 | 12 | 360 |
| 17, 31 | 1 | 527 |