With fractions, the GCD is for simplifying and the LCM is for adding or comparing. For 18 and 24, the GCD is 6 and the LCM is 72: 18/24 reduces to 3/4, and to add 1/18 + 1/24 you convert everything to seventy-seconds. (The GCD is also called the GCF, or HCF in some places.)
| Your question | Tool | With 18 and 24 |
|---|---|---|
| What number do I divide the top and bottom by to simplify? | Greatest common divisor | GCD(18, 24) = 6 |
| Which denominator do I use to add or compare? | Least common denominator | 72, with 4/72 and 3/72 |
| What is the smallest multiple they share? | Least common multiple | LCM(18, 24) = 72 |
| Which primes is each number made of? | Prime factorization | 18 = 2 × 3² and 24 = 2³ × 3 |
| Which numbers divide it exactly? | Divisors | 24 has 8: 1, 2, 3, 4, 6, 8, 12, and 24 |
Read the table by the question you have. The GCD and the LCM come from the same pair of numbers, but they answer different questions: one looks for a common divisor and the other for a common multiple. If you already have the operation written out, the fraction calculator finds the common denominator for you and shows the steps.
One pair of numbers, two jobs
Worked example: GCD and LCM of 18 and 24
You want to reduce 18/24 and, separately, add 1/18 + 1/24. Which number do you need in each case?
- Break each number into primes. 18 = 2 × 3² and 24 = 2³ × 3.
- Take the common factors with the lowest exponent. 2 × 3 = 6, which is the GCD.
- Take all the factors with the highest exponent. 2³ × 3² = 72, which is the LCM.
- Reduce with the GCD. 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so 18/24 = 3/4.
- Add with the LCM. 72 ÷ 18 = 4 and 72 ÷ 24 = 3, so 1/18 = 4/72 and 1/24 = 3/72.
- Add the numerators: 4/72 + 3/72 = 7/72, which can’t be reduced any further.
Result: GCD = 6, LCM = 72, 18/24 = 3/4, and 1/18 + 1/24 = 7/72.
Check: for two positive integers, GCD × LCM equals the product of the two numbers: 6 × 72 = 432 = 18 × 24.
That equality lets you find one of the two values if you know the other. To see the calculation with successive division instead of primes, look at the Euclidean algorithm: 24 = 1 × 18 + 6 and 18 = 3 × 6, so the last nonzero remainder is 6.
Different denominators and cross multiplication
Two fractions with different denominators can be added using any common denominator. Cross multiplication uses the product of the denominators: 18 × 24 = 432, so 1/18 + 1/24 = (24 + 18)/432 = 42/432. The value is the same, but the numbers are six times larger and you have to reduce at the end: 42/432 = 7/72. The LCM saves you that last step because it is already the smallest possible denominator. To decide which of two fractions is larger, though, cross multiplication is handy, and it is explained in comparing fractions.
Something similar happens with building up and simplifying. To build up an equivalent fraction, you multiply the numerator and denominator by the same number; to simplify, you divide both by the same number. Both give equivalent fractions; the difference is the direction. Going from 3/4 to 72/96 means multiplying by 24, and the GCD tells you the most you can divide by when simplifying. More cases in equivalent fractions and simplifying fractions.
Where to go next
Each tool in the table handles one number. If you want to decide between them more carefully or practice, these pages take you further:
- GCD vs. LCM: a guide to choosing in word problems that don’t say which of the two to use.
- Divisibility rules: rules for 2, 3, 5, 9, and 11 to tell whether a number divides evenly without doing the division, useful before looking for the GCD.
- GCD and LCM word problems: sharing into equal groups and events that coincide at regular intervals, with the choice explained in each case.
- Fraction operations: the calculators that use these results to add, subtract, compare, and simplify.
- Adding fractions: the common denominator step applied to a complete sum, with different types of cases.
The divisors tool only accepts positive integers, because zero has infinitely many divisors and they can’t be listed.