Question

Which of the following integers are multiples of both 2 and 3?
Indicate {all such integers.}

• A. 8
• B. 9
• C. 12
• D. 18
• E. 21

Hint:

To quickly check if a number is divisible by 6, use the divisibility rules for 2 and 3. The number must be even (divisible by 2) AND the sum of its digits must be divisible by 3. For example, for 36: it's even, and 3+6=9, which is divisible by 3.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks us to identify all numbers from the list that are multiples of both 2 and 3. An integer that is a multiple of two or more numbers is a multiple of their least common multiple (LCM).
Step 2: Key Approach:
The least common multiple of 2 and 3 is \(2 \times 3 = 6\). Therefore, any integer that is a multiple of both 2 and 3 must be a multiple of 6. The task simplifies to checking which of the given numbers are divisible by 6.
Step 3: Detailed Explanation:
We will test each option for divisibility by 6.
- (A) 8: \(8 \div 6\) does not yield an integer. Not a multiple of 6.
- (B) 9: 9 is a multiple of 3 but not 2. Not a multiple of 6.
- (C) 12: \(12 \div 6 = 2\). 12 is a multiple of 6. This is a correct answer.
- (D) 18: \(18 \div 6 = 3\). 18 is a multiple of 6. This is a correct answer.
- (E) 21: 21 is a multiple of 3 but not 2. Not a multiple of 6.
- (F) 36: \(36 \div 6 = 6\). 36 is a multiple of 6. This is a correct answer.
Step 4: Final Answer:
The integers from the list that are multiples of both 2 and 3 (and therefore of 6) are 12, 18, and 36. The correct options are (C), (D), and (F).