Question

What is the greatest common divisor (GCD) of 36 and 60?

• A. 6
• B. 12
• C. 18
• D. 24

Hint:

For multiple-choice questions, a quick strategy is to test the answer choices, starting with the largest. (D) 24: Does 24 divide 36? No. (C) 18: Does 18 divide 60? No. (B) 12: Does 12 divide 36? Yes (36/12 = 3). Does 12 divide 60? Yes (60/12 = 5). Since 12 works and it's the largest option we've found to work, it must be the GCD.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The greatest common divisor (GCD) of two integers is the largest positive integer that divides both of them without leaving a remainder.
Step 2: Key Formula or Approach:
There are two common methods: listing the factors or using prime factorization.
Step 3: Detailed Explanation:
Method 1: Listing Factors
- Find all the positive divisors (factors) of 36: \{1, 2, 3, 4, 6, 9, 12, 18, 36\}
- Find all the positive divisors (factors) of 60: \{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}
- The common divisors are \{1, 2, 3, 4, 6, 12\}.
- The greatest among these common divisors is 12.
Method 2: Prime Factorization
- Find the prime factorization of 36: \( 36 = 2 \times 18 = 2 \times 2 \times 9 = 2^2 \times 3^2 \)
- Find the prime factorization of 60: \( 60 = 2 \times 30 = 2 \times 2 \times 15 = 2^2 \times 3 \times 5 \)
- To find the GCD, take the lowest power of each common prime factor and multiply them. The common prime factors are 2 and 3.
- Lowest power of 2 is \( 2^2 \).
- Lowest power of 3 is \( 3^1 \).
- GCD = \( 2^2 \times 3^1 = 4 \times 3 = 12 \).
Step 4: Final Answer:
The greatest common divisor of 36 and 60 is 12.