greatest common factor
Suppose you have 12 red tiles and 18 blue tiles and want to lay them in equal-sized square blocks with no tiles left over. The largest block size that divides both counts evenly is 6. That number — the biggest whole number that goes into every given number without remainder — is the greatest common factor.
One reliable way to find it: list the factors of each number and pick the largest they share. The factors of 12 are 1, 2, 3, 4, 6, 12; the factors of 18 are 1, 2, 3, 6, 9, 18; the greatest shared one is 6. A faster way for larger numbers is to use prime factorization and multiply together the primes the numbers have in common.
The greatest common factor is the engine behind reducing fractions to lowest terms and behind factoring out a common piece from an algebraic expression. If two numbers share no factor beyond 1, their greatest common factor is 1, and they are called relatively prime; 8 and 15 are an example.
Find the GCF of 24 and 36. Primes: 24 = 2^3 × 3 and 36 = 2^2 × 3^2. Common part: 2^2 × 3 = 12. So GCF(24, 36) = 12.
Take each shared prime to its smallest power, then multiply to get the GCF.
For large numbers, listing factors is slow; the Euclidean algorithm finds the greatest common factor quickly by repeated division, never needing to know any prime factorization at all.