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5.2. COUNTING FACTORS, GREATEST COMMON FACTOR, AND LEAST COMMON MULTIPLE 83 Set Intersection Method Example. Find GCF(36, 48). A = {x|x is a factor of 36} = {1, 2, 3, 4, 6, 9, 12, 18, 36} B = {x|x is a factor of 48} = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48} A \ B = {x|x is a factor of 36 and 48} = {1, 2, 3, 4, 6, 12} Thus GCF(36, 48) = 12, the largest element in A \ B. Prime Factorization Method Example. Find GCF(36, 60). (1) Factor each number into primes. 36 = 22 · 32 60 = 22 · 3 · 5 (2) For the GCF, take each factor the fewest number of times it is used in any one number. GCF(36, 60) = 22 · 3 = 12. Euclidean Algorithm Method Suppose a and b are whole numbers with a If c | a and c | b, c | a Also, if c | b and c | a of a and b. b. b, so c is a common factor of b and a b, c | [(a b. b) + b)], so c | a and c is a common factor We have proven the following theorem: