GCF and LCM — How to Find Greatest Common Factor & Least Common Multiple
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1. Greatest Common Factor (GCF)
The GCF (also called GCD — Greatest Common Divisor) of two or more numbers is the largest number that divides all of them evenly.
Method 1 — List all 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 factors = {1, 2, 3, 4, 6, 12}. GCF = 12.
Method 2 — Prime factorization: 24 = 2³ × 3. 36 = 2² × 3². GCF = take the lowest power of shared primes = 2² × 3 = 4 × 3 = 12.
2. The Euclidean Algorithm
The most efficient method for finding GCF, especially for large numbers. Algorithm: GCF(a, b) = GCF(b, a mod b), repeat until remainder = 0.
GCF(48, 18): 48 mod 18 = 12. GCF(18, 12): 18 mod 12 = 6. GCF(12, 6): 12 mod 6 = 0. GCF = 6.
3. Least Common Multiple (LCM)
The LCM of two or more numbers is the smallest number that is a multiple of all of them.
Method 1 — List multiples: Multiples of 4 = {4, 8, 12, 16, 20, 24...}. Multiples of 6 = {6, 12, 18, 24...}. LCM = 12.
Method 2 — Prime factorization: 4 = 2². 6 = 2 × 3. LCM = take the highest power of each prime = 2² × 3 = 12.
Method 3 — Use the formula: LCM(a, b) = (a × b) ÷ GCF(a, b). LCM(4, 6) = (4 × 6) ÷ 2 = 24 ÷ 2 = 12.
4. When to Use GCF vs LCM
- Use GCF to simplify fractions, split items into equal groups, or find the common factor.
- Use LCM to add or subtract fractions with different denominators (finding the LCD), synchronise repeating events.