GCF and LCM

The greatest common factor (GCF) of two or more integers is the largest number that divides evenly into all of them. The least common multiple (LCM) is the smallest positive number that all of them divide into evenly. For 12 and 18: the factors they share are 1, 2, 3, and 6, so the GCF is 6; the multiples of 18 are 18, 36, 54, ..., and 36 is the first one that's also a multiple of 12, so the LCM is 36.

Prime factorization handles bigger numbers cleanly. Write each number as a product of primes: 12 = 2² × 3 and 18 = 2 × 3². The GCF takes the lowest power of each shared prime (2¹ × 3¹ = 6), while the LCM takes the highest power of every prime that appears (2² × 3² = 36). A handy check for two numbers: GCF × LCM = the product of the numbers themselves (6 × 36 = 12 × 18 = 216).

Keeping the two straight is half the battle. The GCF is always less than or equal to the smaller number — it's a factor. The LCM is always greater than or equal to the larger number — it's a multiple. Word problems signal which one you need: splitting items into equal groups or simplifying a fraction points to the GCF; finding when repeating events line up again (buses arriving every 12 and 18 minutes coincide every 36) or getting a common denominator points to the LCM.

GCF and LCM questions appear on the ACT and GRE quantitative sections and underpin harder number theory problems on math competitions like the AMC. Achievable's courses for all three walk through the prime factorization method and the word-problem patterns that tell you which of the two to compute.