Identify the largest positive integer that divides two whole numbers evenly. The result panel keeps greatest common factor and its numerical trail together.
Trace Greatest Common Factor back through First integer and Second integer. Those entries should support the displayed Greatest common factor. For a sensitivity check, alter Second integer only. Compare that Greatest Common Factor result with the first Greatest common factor, keeping both cases visible.
For 84 and 126, prime factorization gives 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. Their shared part is 2 × 3 × 7 = 42.
Factors and multiples point in opposite directions: a factor divides a number, while a multiple is produced by multiplying it. The GCF of coprime integers is 1, not 0.
The answer can change abruptly when one integer gains or loses a prime factor. Multiplying both inputs by the same whole number multiplies their GCF by that number. That behavior gives the greatest common factor output a built-in reasonableness test.
Choose GCF when dividing quantities into the largest identical groups. Choose LCM when asking when cycles meet or which smallest denominator or multiple can contain both inputs. Writing “Greatest common factor” beside the output prevents that mix-up.
A GCF simplifies fractions, splits supplies into identical groups, and factors algebraic expressions. If 84 red tiles and 126 blue tiles must form the greatest possible number of identical sets, the GCF gives the set count. A related application of Greatest Common Factor is prime factors.
The greatest common factor, also called the greatest common divisor, is the largest positive integer shared by two integer factor lists. It captures every prime factor the numbers hold in common, using the smaller exponent where a prime repeats. For a connected concept in Greatest Common Factor, see fraction arithmetic.
The Euclidean algorithm is usually faster than listing factors. Divide the larger integer by the smaller, replace the pair with the smaller number and the remainder, and repeat until the remainder becomes zero. A hand-worked extension of Greatest Common Factor is least common multiple.
No. A common factor cannot be larger than the smaller nonzero absolute value.
The GCF of 0 and a nonzero integer is the absolute value of that integer.
Divisibility is unchanged by sign, and the conventional GCF is nonnegative.
Yes. The names greatest common factor and greatest common divisor refer to the same value.