What is the greatest common divisor (GCD) of 48 and 36?

Question

Accepted Answer

12. To find the GCD (also called the greatest common factor or highest common factor) of 48 and 36, we can use prime factorization. First, find the prime factorization of each number: 48 = 2^4 × 3^1 and 36 = 2^2 × 3^2. The GCD is the product of the lowest power of each common prime factor: GCD = 2^2 × 3^1 = 4 × 3 = 12. We can verify: 48 ÷ 12 = 4 and 36 ÷ 12 = 3, and there is no larger number that divides both 48 and 36 evenly.

Another method is to list all the divisors of each number. Divisors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. The common divisors are: 1, 2, 3, 4, 6, 12. The greatest of these is 12. A third efficient method is the Euclidean algorithm: 48 = 1(36) + 12, then 36 = 3(12) + 0. Since the remainder is 0, the GCD is the last nonzero remainder, which is 12.

The GCD is important in simplifying fractions (dividing numerator and denominator by their GCD), solving problems involving divisibility, and various applications in number theory. A related concept is the least common multiple (LCM), which can be found using the relationship LCM(a,b) = (a × b) / GCD(a,b). For 48 and 36: LCM = (48 × 36) / 12 = 1728 / 12 = 144. Both GCD and LCM are commonly tested topics on the CLEP College Mathematics exam.

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