Common Factor Calculator

This common factor calculator takes two whole numbers and shows you everything about how they divide: the complete list of factors for each number, the factors they share, the single Greatest Common Factor (GCF), and the Least Common Multiple (LCM). Finding common factors is one of the foundational skills in arithmetic and number theory, and it turns up constantly — in simplifying fractions, solving equations, working with ratios and proportions, and in practical scheduling problems like working out when two repeating events line up again. It is built for students learning factors and multiples, parents helping with homework, and anyone who needs a quick, reliable answer without doing the long division by hand.

What are factors?

A factor (or divisor) of a number is any whole number that divides into it exactly, with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12, because each of those divides 12 evenly. Every number has at least two factors — 1 and itself — and the factors of a number always come in pairs that multiply back to it (for 12: 1×12, 2×6, 3×4). The calculator finds these efficiently by testing divisors only up to the square root and pairing each one with its partner.

Listing the common factors

The common factors of two numbers are the values that appear in both of their factor lists. Take 12 and 18: the factors of 12 are 1, 2, 3, 4, 6, 12 and the factors of 18 are 1, 2, 3, 6, 9, 18. The numbers in both lists — 1, 2, 3 and 6 — are the common factors. Every pair of whole numbers shares at least the factor 1.

Greatest Common Factor (GCF)

The GCF — also called GCD (Greatest Common Divisor) or HCF (Highest Common Factor) — is simply the largest of those common factors. For 12 and 18 the common factors are 1, 2, 3 and 6, so the GCF is 6. The calculator finds it with the fast Euclidean algorithm: gcd(a, b) = gcd(b, a mod b), repeated until the remainder reaches zero.

Least Common Multiple (LCM)

The LCM is the smallest positive number that both values divide into evenly — the flip side of the GCF. It is computed as LCM(a, b) = a × b ÷ GCF(a, b). For 12 and 18 that is 12 × 18 ÷ 6 = 216 ÷ 6 = 36. The LCM is what you need as a common denominator when adding or subtracting fractions.

Worked example

Suppose you want to simplify the fraction 12/18. List the factors: 12 gives 1, 2, 3, 4, 6, 12 and 18 gives 1, 2, 3, 6, 9, 18. The shared factors are 1, 2, 3, 6, and the greatest is 6, so GCF = 6. Divide top and bottom by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3, giving the fully reduced fraction 2/3. If you instead needed a common denominator for adding 1/12 and 1/18, you would use the LCM, 36.

Tips

• To simplify any fraction in one step, divide the numerator and denominator by their GCF.
• If the GCF comes out as 1, the numbers are coprime and the fraction is already in lowest terms.
• For three or more numbers, find the GCF of two, then take the GCF of that result with the next number.
• Use the LCM, not the simple product of the denominators, to keep fraction arithmetic from blowing up into large numbers.

Common mistakes

• Confusing GCF with LCM — the GCF is the largest factor they share, the LCM is the smallest multiple they share.
• Forgetting that 1 is a factor of every number, so the common-factor list is never empty.
• Stopping at any common factor instead of the greatest one when simplifying — that leaves the fraction only partly reduced.
• Treating a prime-looking pair as having no common factor; even 8 and 15 share the factor 1, which simply means they are coprime.