What Are GCF and LCM?
Two tools that help you simplify fractions, find common denominators, and solve real-world problems.
Factors and Multiples — Quick Review
Before diving into GCF and LCM, let's clarify two terms. A factor of a number divides into it evenly with no remainder. The factors of 12 are: 1, 2, 3, 4, 6, and 12. A multiple of a number is what you get when you multiply it by whole numbers. The multiples of 4 are: 4, 8, 12, 16, 20, 24… (the list goes on forever).
GCF — Greatest Common Factor
The Greatest Common Factor is the largest number that divides evenly into two or more numbers. To find the GCF of 18 and 24: list the factors of each. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Common factors: 1, 2, 3, 6. The greatest is 6.
GCF is essential for simplifying fractions. To simplify 18/24, divide both numerator and denominator by the GCF (6): 18÷6 / 24÷6 = 3/4. Done in one step.
LCM — Least Common Multiple
The Least Common Multiple is the smallest number that is a multiple of two or more numbers. To find the LCM of 6 and 8: list multiples. Multiples of 6: 6, 12, 18, 24, 30… Multiples of 8: 8, 16, 24, 32… The first number that appears in both lists is 24.
LCM is essential for adding fractions with different denominators. To add 1/6 + 1/8, you need a common denominator — the LCM of 6 and 8, which is 24. Convert: 4/24 + 3/24 = 7/24.
Real-World Uses
GCF in action: You have 18 red flowers and 24 yellow flowers and want to make identical bouquets with no flowers left over. The GCF (6) tells you the maximum number of bouquets: 6 bouquets with 3 red and 4 yellow each. LCM in action: Bus A comes every 6 minutes and Bus B comes every 8 minutes. If both arrive at 8:00 AM, when will they next arrive together? In 24 minutes — at 8:24 AM.
The Prime Factorization Shortcut
For larger numbers, listing all factors or multiples is tedious. A faster method uses prime factorization. Break each number into its prime factors: 18 = 2 × 3 × 3, and 24 = 2 × 2 × 2 × 3. For GCF, take the smallest power of each shared prime: 2¹ × 3¹ = 6. For LCM, take the largest power of every prime that appears: 2³ × 3² = 72. This method works for any numbers, no matter how large.
GCF and LCM have been used for thousands of years. Ancient Egyptian mathematicians used concepts similar to GCF to simplify fractions on papyrus scrolls around 1650 BCE. The systematic method for finding GCF that we learn today — called the Euclidean algorithm — was described by the Greek mathematician Euclid around 300 BCE in his famous work "Elements." It's one of the oldest algorithms still in everyday use, predating modern computers by over 2,200 years.
Last reviewed: April 2026