GCF Calculator

Before you can simplify a fraction, split items into equal groups with nothing left over, or tile a floor with the largest possible squares, you need one number: the greatest common factor. The GCF calculator finds the largest whole number that divides two or more numbers exactly, sparing you the factor-listing grind — especially valuable when the numbers run into the hundreds and their factor lists run long.

Quick Answer

The GCF is the largest factor shared by all the given numbers. For 48 and 180, the shared factors top out at 12. One reliable route: break each number into primes — 48 = 2⁴ × 3 and 180 = 2² × 3² × 5 — then multiply the primes they share, each taken at the lower power: 2² × 3 = 12.

Three Roads to the Same Answer

The listing method is the most transparent. Write out every factor of each number and pick the largest entry on both lists. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180. Scanning the overlap, the biggest shared name is 12. Listing is unbeatable for understanding, but it scales poorly — miss one factor of a large number and the answer is wrong with total confidence.

The prime factorisation method trades listing for structure. Express each number as a product of primes, line up the primes the numbers have in common, and take each common prime as many times as it appears in both factorisations (the smaller count governs). This is the method to trust for three or more numbers, and it explains a useful edge case: if the numbers share no primes at all, their GCF is 1 and they are called coprime, like 17 and 23.

The third road, the Euclidean algorithm, is what computers use. Divide the larger number by the smaller, keep the remainder, and repeat — 180 ÷ 48 leaves 36, 48 ÷ 36 leaves 12, 36 ÷ 12 leaves 0 — and the last nonzero remainder, 12, is the GCF. Whichever road you travel, the destination is identical, which makes one method a perfect check on another.

How to Use the GCF Calculator

1. Enter the first number as a positive whole number.
2. Enter the second number; many GCF tools accept a third or more, and the factor must divide every number entered.
3. Read the greatest common factor.
4. Verify quickly: divide each of your numbers by the result — both divisions should come out whole, with nothing left over.
5. If you are simplifying a fraction, divide its top and bottom by the GCF in one step to reach lowest terms.

Simplifying 48/180 All the Way Down

Watch the GCF earn its living. The fraction 48/180 looks unwieldy, but both numbers are divisible by 12, the GCF computed above. Dividing through: 48 ÷ 12 = 4 and 180 ÷ 12 = 15, giving 4/15 — fully simplified in a single move, since 4 and 15 share no factor beyond 1. Without the GCF you might divide by 2, then 2 again, then 3, arriving eventually but with three chances to slip.

A small reference set:

• GCF(24, 36) = 12
• GCF(56, 98) = 14, from 56 = 2³ × 7 and 98 = 2 × 7² sharing 2 × 7
• GCF(17, 23) = 1 — two primes are always coprime to each other
• GCF(100, 100) = 100 — a number is its own greatest factor

GCF Confusions and How to Dodge Them

The number-one confusion is with the LCM, the least common multiple. They are near-opposites: the GCF is a divisor, never larger than the smallest number you entered, while the LCM is a multiple, never smaller than the largest. If your "GCF" of 48 and 180 comes out as 720, you have computed the LCM by accident. The second trap is stopping at a common factor instead of the greatest — spotting that both numbers are even and answering 2, when 12 was available. Any common factor is useful for one simplifying step; only the GCF finishes the job at once.

With three numbers, the factor must divide all three, and it can drop sharply: GCF(48, 180, 24) is still 12, but add 35 to the set and the GCF collapses to 1. Also note the special cases rather than fearing them: the GCF of any number and 1 is 1, and if the smaller number divides the larger exactly, the smaller number is the GCF — GCF(18, 54) = 18, no calculation needed.

The Largest Square Tile Problem

Here is the GCF doing manual labour. You are tiling a rectangular floor 48 cm by 180 cm with the largest possible square tiles, cutting none. The tile side must divide both dimensions exactly, and the largest such side is GCF(48, 180) = 12 cm. Along the width, 48 ÷ 12 = 4 tiles fit; along the length, 180 ÷ 12 = 15 tiles; the floor takes 4 × 15 = 60 tiles, with zero waste and zero cuts. The same reasoning sizes the largest equal groups in any fair division: 48 juniors and 180 seniors forming teams with the same club mix can make at most 12 identical groups. Whenever a problem asks for the biggest equal unit that fits two quantities exactly — ribbons from two spools, rows in two gardens, packets from two stocks — it is quietly asking for a greatest common factor.

Three-number problems bow to the same prime method. Take 36, 60 and 84: factorising gives 2² × 3², 2² × 3 × 5, and 2² × 3 × 7. The primes present in all three are 2 and 3, at lowest powers 2² and 3¹, so GCF(36, 60, 84) = 4 × 3 = 12 — each additional number can only hold the GCF steady or pull it down, never raise it.

Factoring's Neighbourhood

GCF work is one branch of the divisibility family. The factor calculator lists every factor of a single number, the remainder calculator shows what is left when division does not come out even, and the long division calculator lays out the full division process step by step. Explore them together in the math calculators section.