Common Factor Calculator

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides two or more integers without leaving a remainder. For example, the GCF of 24 and 36 is 12, because 12 is the largest number that divides both 24 and 36 without a remainder.

The least common multiple (LCM) is the smallest positive integer that is divisible by two or more integers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that is divisible by both 4 and 6. The LCM is useful when working with fractions or finding common denominators.

The Euclidean algorithm is an efficient method for finding the GCF of two numbers. It works by repeatedly dividing the larger number by the smaller number and taking the remainder, then dividing the smaller number by this remainder, and so on, until the remainder is zero. The last non-zero remainder is the GCF. For example, to find the GCF of 24 and 36: 36 ÷ 24 = 1 remainder 12; 24 ÷ 12 = 2 remainder 0; so the GCF is 12.

To find the GCF using prime factorization, you first break down each number into its prime factors. Then, you identify the common prime factors and multiply them together. For example, to find the GCF of 24 and 36: 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3. The common prime factors are 2, 2, and 3, so the GCF is 2 × 2 × 3 = 12.

For any two positive integers a and b, the product of their GCF and LCM equals the product of the numbers themselves: GCF(a, b) × LCM(a, b) = a × b. This relationship can be useful when you know one value and need to find the other. For example, if the GCF of 24 and 36 is 12, then the LCM is (24 × 36) ÷ 12 = 72.