Greatest Common Factor Calculator

GCF, GCD, and HCF name the same integer result

Greatest common factor (GCF), greatest common divisor (GCD), and highest common factor (HCF) are different names for the largest positive integer that divides every integer in a list without a remainder.

For 84, 126, and 210, the result is 42. Each value is a multiple of 42: 84 = 42 × 2, 126 = 42 × 3, and 210 = 42 × 5. No larger positive integer divides all three.

How the Euclidean algorithm works

For two positive integers, divide the larger by the smaller and record the remainder. Replace the pair with the previous divisor and that remainder, then repeat. The last nonzero divisor is the GCF.

For 84 and 126, the visible equations are 126 = 84 × 1 + 42 and 84 = 42 × 2 + 0. The last nonzero divisor is 42, so GCF(84, 126) = 42.

How more than two integers are merged

The calculator folds through the list from left to right. It finds the GCF of the first pair, then combines that result with the next integer. For the worked example, the first merge gives 42; the second merge calculates GCF(42, 210), which also gives 42.

Every merge opens into its own remainder chain. This makes the multi-number calculation auditable without switching to a separate two-number Euclidean tool.

Negative integers and zero

Signs do not change positive divisibility, so negative inputs are normalized to absolute values before calculation. For example, −24, 36, and 60 are shown as 24, 36, and 60, with a GCF of 12.

Zero can appear with a nonzero integer: GCF(0, n) = |n|. An all-zero list is different. Every positive integer divides zero, so there is no greatest positive common factor; the calculator asks for at least one nonzero value.

Input limits and exactness

Enter 2–20 base-10 whole integers separated by spaces, commas, or new lines. Each absolute integer may contain at most 15 digits. A leading minus sign is accepted; decimals, fractions, plus signs, exponent notation, symbols, and polynomial variables are rejected.

The calculation uses JavaScript BigInt integer division and remainder operations. It does not convert values to floating point or round the result. The limits keep the page and the full step receipt responsive on a phone.

What this calculator does not do

This page does not use prime factorization and does not list every factor. It also does not calculate LCM, factor fractions, or solve the GCF of monomials, polynomials, variables, or symbolic expressions. Those are separate tasks with different inputs and explanations.