What is the greatest common factor (GCF) of 24 and 36?

GCF (greatest common factor) is the largest number that divides both. Method 1 — prime factorization: 24 = 2³ × 3; 36 = 2² × 3². Common prime factors with smallest exponents: 2² (smaller of 2³ and 2²) and 3¹ (smaller of 3¹ and 3²). GCF = 2² × 3 = 4 × 3 = 12. Method 2 — list factors: factors of 24: 1, 2, 3, 4, 6, 8, 12, 24; factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36; common: 1, 2, 3, 4, 6, 12; greatest is 12. LCM (least common multiple) — smallest number divisible by both: use prime factorization with LARGEST exponents: LCM(24, 36) = 2³ × 3² = 8 × 9 = 72. Relationship: GCF × LCM = product of the numbers. Check: 12 × 72 = 864 = 24 × 36 ✓. Applications: reducing fractions (divide by GCF); adding fractions (LCM as common denominator); cycling problems; tile patterns; arrangement problems.