What is the greatest common divisor of 48 and 180?

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Multiple Choice

What is the greatest common divisor of 48 and 180?

Explanation:
The greatest common divisor is the largest number that divides both numbers exactly. Break each number into prime factors: 48 = 2^4 × 3 and 180 = 2^2 × 3^2 × 5. The common primes are 2 and 3. For the gcd, use each common prime to the smallest power it has in either factorization: 2^2 (which is 4) and 3^1 (which is 3). Multiply them together: 4 × 3 = 12. So the greatest common divisor is 12. You can also see this quickly with the Euclidean algorithm: 180 mod 48 = 36, then 48 mod 36 = 12, then 36 mod 12 = 0, confirming gcd is 12.

The greatest common divisor is the largest number that divides both numbers exactly. Break each number into prime factors: 48 = 2^4 × 3 and 180 = 2^2 × 3^2 × 5. The common primes are 2 and 3. For the gcd, use each common prime to the smallest power it has in either factorization: 2^2 (which is 4) and 3^1 (which is 3). Multiply them together: 4 × 3 = 12. So the greatest common divisor is 12. You can also see this quickly with the Euclidean algorithm: 180 mod 48 = 36, then 48 mod 36 = 12, then 36 mod 12 = 0, confirming gcd is 12.

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