The greatest common factor (GCF, also called the greatest common divisor, GCD) is for splitting: in 18/24, the GCF of 18 and 24 is 6, and it takes the fraction to 3/4. The least common multiple (LCM) is for joining: to add two fractions with denominators 4 and 6, LCM(4, 6) = 12 gives the common denominator. If you need the hand method for the GCF, see the Euclidean algorithm.
Before you calculate, ask yourself one question. Do you want to split a quantity into smaller parts, or join two quantities into one common measure? If you split, it is the GCF. If you join, it is the LCM. To practice that decision, try the GCD and LCM word problems.
| Criterion | GCF | LCM |
|---|---|---|
| What it finds | The greatest common factor | The least common multiple |
| Which numbers it looks at | The numerator and denominator of one fraction | The denominators of several fractions |
| What it is for | Simplifying | Adding, subtracting and comparing |
| Example | GCF(18, 24) = 6, which reduces 18/24 in one step | LCM(4, 6) = 12, and the sum moves to twelfths |
The table sums up the guide. The GCF works inside a fraction, and the LCM works between fractions.
The GCF works inside the fraction
Simplifying means dividing the numerator and the denominator by the same number, and the GCF does it in a single step. The common factors of 18 and 24 are 1, 2, 3 and 6. If you divide by 2 you get 9/12, which can still be reduced. With the GCF, 6, you go straight to the result, which is already in lowest terms because 3 and 4 share only 1.
For large numbers, the Euclidean algorithm is faster than listing factors. To apply it to a fraction, use simplify fractions, and to calculate just the number, use the greatest common divisor calculator.
Worked example: juice and water
You mix 3/4 of a liter of juice with 5/6 of a liter of water. How many liters do you have?
- Look at the denominators. They are 4 and 6, and you need to join them, so use the LCM.
- Find the LCM. The multiples of 4 are 4, 8, 12… and those of 6 are 6, 12… The first one they share is 12.
- Build up. 3/4 = 9/12 and 5/6 = 10/12.
- Add. 9/12 + 10/12 = 19/12, and since 19 and 12 share no factor, there is nothing left to reduce with the GCF.
You have 19/12 of a liter, which is 1 7/12.
Check. In decimals, 0.75 + 0.83… = 1.58…, and 19 ÷ 12 also gives 1.58….
The LCM works between fractions
Two fractions can be added or subtracted only when they count parts of the same size. If they have different denominators (unlike fractions), you have to rewrite them with a denominator that is a multiple of both, and the smallest possible one is the LCM. If they already share a denominator (like fractions), you do not need the LCM: you add the numerators, as the guide on why you do not add denominators explains.
The same common denominator works for comparing. In twelfths, 9/12 is less than 10/12, so the first fraction is the smaller one. Cross multiplication gives the same answer: 3 × 6 = 18 is less than 5 × 4 = 20. It is explained in compare fractions. To see the common denominator already worked out, use the least common denominator tool or the least common multiple calculator.
Building up and simplifying are opposite operations
To build up a fraction, you multiply its numerator and denominator by the same number, which is what you do when you move to the common denominator. Simplifying is dividing by the same number, which is what the GCF does. Both keep the value of the fraction and produce equivalent fractions, a topic covered in equivalent fractions.
In a sum, both often appear in order: first the LCM to add, and then, if needed, the GCF to simplify. With 1/6 + 1/3, the LCM of 6 and 3 is 6, the sum is 1/6 + 2/6 = 3/6, and the GCF of 3 and 6 is 3, which takes the result to 1/2.
How the two numbers are related
For two positive numbers, GCF × LCM = the product of the two. With 4 and 6 it is 2 × 12 = 24 = 4 × 6, which is why you can find the LCM as the product divided by the GCF. This relation holds only for two numbers.
| Numbers | GCF | LCM | GCF × LCM |
|---|---|---|---|
| 4 and 6 | 2 | 12 | 24 |
| 6 and 8 | 2 | 24 | 48 |
| 4 and 9 | 1 | 36 | 36 |
| 12 and 18 | 6 | 36 | 216 |
In every row, the last value matches the product of the two numbers: 6 × 8 = 48, 4 × 9 = 36 and 12 × 18 = 216.
How to choose in a word problem
Keywords can mislead you, so translate the question. If it asks for the largest size that fits a whole number of times into two measurements, it is the GCF, as when you cut equal pieces with nothing left over. If it asks when two things that repeat at regular intervals line up, it is the LCM. For practice with unlabeled problems, see the GCD and LCM word problems.
Sources
OpenStax (Prealgebra 2e, section 4.5) defines the least common denominator of two fractions as the LCM of their denominators, which is the section this guide relies on for adding with different denominators: OpenStax, add and subtract fractions with different denominators, accessed October 17, 2026. Math Is Fun (greatest common factor page) defines it as the highest number that divides exactly into two or more numbers, and presents it as the tool for simplifying fractions: Math Is Fun, greatest common factor, accessed October 17, 2026.