Article
Greatest common divisor
The GCD is the largest integer that divides both values with no remainder. It is what you use to shorten a fraction.
The GCD calculator does this.
The idea in brief
The GCD is the largest integer that divides both values with no remainder. You look for a shared divisor and keep the greatest one.
48 and 18 both divide by 6. They are not both divisible by 12, because 18 / 12 leaves a remainder. So the GCD is 6, not 12.
On this page: GCD is the greatest common divisor: the largest integer that divides both values with no remainder. LCM is the least common multiple, a different number. GCD(48, 18) means the GCD of 48 and 18.
When this helps
| Situation | What the card shows |
|---|---|
| You want to shorten a fraction to lowest terms. | You divide the numerator and the denominator by the greatest common divisor. |
| You want to see whether two integers still share a factor. | If the GCD is 1, the integers are coprime and the fraction will not shorten further. |
| You want to check the GCD on paper with remainders. | Divide the larger number by the smaller one and repeat with the remainder until the remainder is zero. |
How to read it
GCD(48, 18) = 6, because 6 divides both and 12 does not divide 18. GCD(17, 13) = 1: the pair is coprime. If one number is zero, the result is the absolute value of the other. Two zeros do not give a useful divisor on this card.
The card takes integers. It is not LCM and not a factor tree printed as a list. If you want to reduce 18/48, the divisor 6 takes the fraction to 3/8.
A worked example
GCD(48, 18)
- 18 goes into 48 twice, remainder 12.
- 12 goes into 18 once, remainder 6.
- 6 goes into 12 evenly. Stop. GCD = 6.
18/48 reduced by 6 is 3/8.
Typical numbers from the card
| Inputs | Result |
|---|---|
| You want the GCD of 48 and 18. | The result is 6. |
| You want the GCD of 17 and 13. | The result is 1. These numbers will not shorten by a common divisor greater than 1. |
| You divide the fraction 18/48 by a GCD of 6. | After shortening, the fraction is 3/8. |
Typical mistakes
| Mistake | What goes wrong | How to avoid it |
|---|---|---|
| You take 12 as the GCD of 48 and 18. | 12 does not divide 18 with no remainder, so it cannot be the GCD. | The largest integer that divides both values is 6. |
| You mix up GCD with LCM. | LCM is a different number: the least common multiple. | LCM is the least common multiple, which is a different number. |
| You type fractions instead of integers. | This card accepts integers. | Write both values as integers first. |
Before you shorten a fraction
- Both values are integers.
- You are looking for a divisor of both numbers, not a shared multiple.
- If the GCD is 1, the fraction will not shorten further.
- Two zeros do not give a meaningful divisor on this card.
Frequently asked questions
What is GCD(48, 18)?
6. That is the largest integer that divides both with no remainder.
What does GCD = 1 mean?
The numbers are coprime. Example: 17 and 13.
What if one number is 0?
The result is the absolute value of the other.
Is this LCM?
No. LCM is the least common multiple, a different value.
How does this help fractions?
Divide numerator and denominator by the GCD. 18/48 after 6 is 3/8.
Will fractions in the inputs work?
The card is for integers. Write them as integers first.