GCF + LCM Calculator

About GCF and LCM

The greatest common factor (GCF) of a set of whole numbers is the largest number that divides all of them with no remainder. The least common multiple (LCM) is the smallest positive number that every one of them divides into. They are two of the most useful ideas in elementary number theory: the GCF answers "how large a shared piece can I break these into?" and the LCM answers "when do these cycles line up?". This calculator takes a comma-separated list and returns both, along with their product, for two or more values.

GCF is also called the greatest common divisor (GCD), the term used in mathematics and computing, while GCF is the label taught in school. For exactly two numbers the two quantities are tied together by a clean identity: the GCF multiplied by the LCM equals the product of the two numbers. That relationship is what lets the calculator compute the LCM cheaply once it has the GCF.

How it works

The tool finds the GCF with the Euclidean algorithm, one of the oldest algorithms still in use, then derives the LCM from it:

GCF(a, b): repeat  a, b = b, a mod b  until b = 0; GCF is a
LCM(a, b) = |a x b| / GCF(a, b)
For two numbers:  GCF x LCM = a x b
Many numbers:  fold pairwise, e.g. GCF(a,b,c) = GCF(GCF(a,b), c)

• Euclid is fast: it uses repeated division instead of listing every factor, so it handles large numbers instantly.
• LCM from GCF: once the GCF is known, the LCM is just the product divided by it, avoiding a separate search.
• Coprime check: if the GCF is 1, the numbers share no common factor and are called coprime (relatively prime).

Worked example

Find the GCF and LCM of 12 and 18.

1. Euclidean steps: 18 mod 12 = 6, then 12 mod 6 = 0, so GCF = 6.
2. Product: 12 x 18 = 216.
3. LCM: 216 / 6 = 36.
4. Check the identity: GCF x LCM = 6 x 36 = 216 = 12 x 18. It holds.
5. Use it: the GCF of 6 simplifies 12/18 to 2/3; the LCM of 36 is the common denominator for adding twelfths and eighteenths.

GCF and LCM reference

Using GCF and LCM with fractions

The two operations show up together whenever you work with fractions. To reduce a fraction to lowest terms, divide the numerator and denominator by their GCF: 18/24 has a GCF of 6, so it simplifies to 3/4. To add or subtract fractions, you first need a common denominator, and the smallest one to use is the LCM of the denominators. Adding 1/6 and 1/8, the LCM of 6 and 8 is 24, so the sum becomes 4/24 + 3/24 = 7/24.

Cycle problems are the other classic use. If one bus comes every 6 minutes and another every 8 minutes, they next arrive together after LCM(6, 8) = 24 minutes. If you want to cut two ribbons of 12 cm and 18 cm into equal pieces with none left over, the longest each piece can be is GCF(12, 18) = 6 cm. Recognising which question you are asking, a shared piece (GCF) or a shared cycle (LCM), tells you which output to read.

Common pitfalls

• Swapping the two. The GCF is never larger than the smallest input; the LCM is never smaller than the largest. If your answer breaks that, you have them reversed.
• Assuming LCM equals the product. That is only true when the numbers are coprime (GCF = 1). When they share a factor, the LCM is smaller than the product.
• Forgetting the GCF x LCM identity is for two numbers. It does not extend directly to three or more; fold pairwise instead.
• Mishandling negatives and zero. The GCF is taken on absolute values and the LCM is always positive; the LCM with zero is undefined.
• Listing factors for big numbers. Brute-force factor listing is slow and error-prone; the Euclidean algorithm is the reliable method.

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Frequently asked questions