How to Find GCD and LCM and Use Them to Simplify Fractions
Learn how to find the greatest common divisor and least common multiple, simplify fractions, and add fractions with different denominators.
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What GCD and LCM mean
For 12 and 18, the GCD is 6 and the LCM is 36.
- GCD (greatest common divisor): the largest number that divides all the given numbers exactly.
- LCM (least common multiple): the smallest positive number that all the given numbers divide into exactly.
A quick relationship
For two positive integers a and b, GCD × LCM = a × b. Check with 12 and 18: 6 × 36 = 216, and 12 × 18 = 216. This lets you find the LCM once you know the GCD.
Simplify a fraction with the GCD
Divide the numerator and denominator by their GCD. For 18/24, the GCD is 6, so the simplified fraction is 3/4.
Add fractions with the LCM
To add 1/4 and 1/6, use the LCM of the denominators, 12. Convert to 3/12 and 2/12, then add to get 5/12.
- Find the LCM of the denominators.
- Rewrite each fraction with that denominator.
- Add the numerators and simplify if possible.
FAQ
What is the easiest way to find the GCD?
List the factors of each number and pick the largest shared one, or use the Euclidean algorithm for larger numbers.
Can two numbers have a GCD of 1?
Yes. Such numbers are called coprime, and a fraction made from them is already in simplest form.
Why do I need the LCM for adding fractions?
The LCM gives the smallest common denominator, which keeps the numbers small and the answer easy to simplify.
