What is the greatest common divisor of 630 and 36?

What is the greatest common divisor of 630 and 36?

• A. 6
• B. 9
• C. 18
• D. 36

Answer: (C)

The greatest common divisor is the largest number that divides both numbers. Let’s factor them: 630 = 2 × 3^2 × 5 × 7 and 36 = 2^2 × 3^2. The gcd takes each common prime to the smallest exponent it has in both factorizations, so we get 2^1 and 3^2, which multiply to 2 × 9 = 18. This means 18 divides both 630 and 36, and no larger number does. You can also confirm with the Euclidean approach: 630 mod 36 is 18, then gcd(36,18) = 18. So the greatest common divisor is 18.

The greatest common divisor is the largest number that divides both numbers. Let’s factor them: 630 = 2 × 3^2 × 5 × 7 and 36 = 2^2 × 3^2. The gcd takes each common prime to the smallest exponent it has in both factorizations, so we get 2^1 and 3^2, which multiply to 2 × 9 = 18. This means 18 divides both 630 and 36, and no larger number does. You can also confirm with the Euclidean approach: 630 mod 36 is 18, then gcd(36,18) = 18. So the greatest common divisor is 18.