GCF and LCM Calculator

This GCF and LCM calculator finds the greatest common factor and least common multiple of any set of whole numbers. The greatest common factor and least common multiple are the two workhorses of fraction arithmetic. Enter any set of whole numbers and this calculator returns both, plus the prime factorisation of each input so you can see the shared factors directly.

What they mean

GCF(12, 18) = 6   → the largest number dividing both
LCM(12, 18) = 36  → the smallest number both divide into

For two numbers the identity GCF × LCM = a × b always holds — here 6 × 36 = 216 = 12 × 18.

How they are computed

The GCF uses Euclid's algorithm, repeatedly replacing the larger number by the remainder until one reaches zero. The LCM is then a × b ÷ GCF, extended across the whole list. Prime factorisation is shown for verification.

Worked example

The GCF and LCM of 48 and 180, by Euclid's algorithm:

180 = 3 × 48 + 36
 48 = 1 × 36 + 12
 36 = 3 × 12 + 0    → remainder zero, so GCF = 12
LCM = (48 ÷ 12) × 180 = 4 × 180 = 720

The identity GCF × LCM = a × b holds: 12 × 720 = 8,640 = 48 × 180. The prime factorisations agree as well — 48 = 2⁴ × 3 and 180 = 2² × 3² × 5, so the GCF takes the lowest power of each shared prime (2² × 3 = 12) and the LCM takes the highest (2⁴ × 3² × 5 = 720). Both routes reach the same pair, which is the check worth running when you are doing it on paper.

Why the LCM divides before it multiplies

The obvious formula is a × b ÷ GCF and it is mathematically identical. The implementation computes (a ÷ GCF) × b instead, dividing first. The reason is overflow: for two large numbers the product a × b can exceed what a 64-bit integer holds even when the LCM itself fits comfortably inside it, and the division-first ordering never produces that intermediate value.

The same care runs through the rest of the tool. Everything is exact integer arithmetic, so there is no rounding anywhere and no practical upper limit. Euclid's algorithm is also remarkably quick — its step count grows with the logarithm of the input rather than with the input itself, which is why two fifteen-digit numbers resolve instantly instead of visibly grinding.

What the two numbers are actually for

The LCM is the tool for adding fractions with unlike denominators: 1/6 + 1/8 needs a common denominator, and LCM(6, 8) = 24 is the smallest one that works. It also settles scheduling questions — two events repeating every 6 days and every 8 days coincide every 24 days.

The GCF is the tool for reducing fractions and for dividing things into equal groups. Twelve items and eighteen items can be split into groups of at most six with nothing left over, because GCF(12, 18) = 6. It is also the side length of the largest square that tiles a 12 × 18 rectangle exactly, and it is the basis of every simplification Fraction Calculator performs.

Both extend naturally beyond a pair: GCF(a, b, c) = GCF(GCF(a, b), c), and the LCM the same way, which is how this tool handles a whole list of inputs rather than only two. Ratio Calculator uses the identical GCF step to reduce a ratio to its lowest terms.

Private and instant

All of it runs in your browser with exact integer arithmetic. Nothing is uploaded.