What is the greatest common divisor of 9 and 12?

• A. 3
• B. 1
• C. 2
• D. 4

Answer: (A)

The greatest common divisor is the largest positive integer that divides both numbers without a remainder. For 9 and 12, the divisors of 9 are 1, 3, and 9; the divisors of 12 are 1, 2, 3, 4, 6, and 12. The common divisors are 1 and 3, and the larger one is 3, so the greatest common divisor is 3. You can also see this from prime factorization: 9 = 3^2 and 12 = 2^2 · 3, so the only common prime factor is 3, with the smallest power giving gcd = 3. Numbers like 2 or 4 don’t work because they don’t divide 9, and 1 is common but not the greatest.