LCM & GCF Calculator

Understanding LCM, GCF, and Prime Factorization

The Least Common Multiple (LCM) and Greatest Common Factor (GCF) are essential concepts in number theory and everyday arithmetic. The LCM of two or more numbers is the smallest positive integer that is divisible by each of them. The GCF, also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides each number without a remainder.

Finding the LCM is useful when adding fractions with different denominators: you need the LCD (least common denominator), which is the LCM of the denominators. The GCF helps simplify fractions to their lowest terms by dividing both numerator and denominator by their GCF. Both operations rely on prime factorization, the process of expressing a number as a product of prime numbers.

To compute the GCF using the Euclidean algorithm, repeatedly divide the larger number by the smaller and take the remainder until the remainder is zero. The last non-zero remainder is the GCF. The LCM can then be found using the relationship LCM(a,b) = (a x b) / GCF(a,b). For more than two numbers, apply the operation pairwise. This calculator displays the LCM, GCF, and complete prime factorization for each number you enter, supporting any combination of positive integers.