Find the greatest common divisor of 84 and 120.

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

Find the greatest common divisor of 84 and 120.

Explanation:
To find the greatest common divisor, we look for the largest number that divides both 84 and 120 without a remainder. Using prime factorization helps to see what they share: 84 factors to 2^2 × 3 × 7, and 120 factors to 2^3 × 3 × 5. The common factors are 2 and 3, taken at the smallest powers that appear in both numbers, which is 2^2 and 3. Multiply them: 2^2 × 3 = 4 × 3 = 12. This number divides both 84 (84 ÷ 12 = 7) and 120 (120 ÷ 12 = 10), and you can’t get a larger common divisor because any larger candidate would require a higher power of a shared prime or an additional shared prime that isn’t present in both factorizations. So the greatest common divisor is 12.

To find the greatest common divisor, we look for the largest number that divides both 84 and 120 without a remainder. Using prime factorization helps to see what they share: 84 factors to 2^2 × 3 × 7, and 120 factors to 2^3 × 3 × 5. The common factors are 2 and 3, taken at the smallest powers that appear in both numbers, which is 2^2 and 3. Multiply them: 2^2 × 3 = 4 × 3 = 12. This number divides both 84 (84 ÷ 12 = 7) and 120 (120 ÷ 12 = 10), and you can’t get a larger common divisor because any larger candidate would require a higher power of a shared prime or an additional shared prime that isn’t present in both factorizations. So the greatest common divisor is 12.

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