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
• F. 36

Hint:

Remember the divisibility rules. A number is divisible by 2 if it's even. A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 6 if it is divisible by both 2 and 3.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
An integer that is a multiple of both 2 and 3 must be a multiple of their least common multiple (LCM).
A multiple of 2 is an even number.
A multiple of 3 is a number whose digits sum to a multiple of 3.
Step 2: Key Formula or Approach:
The LCM of 2 and 3 is \( 2 \times 3 = 6 \). Therefore, we are looking for numbers in the list that are multiples of 6.
Step 3: Detailed Explanation:
We will check each option for divisibility by 6 (or by both 2 and 3).


(A) 8: Is a multiple of 2 (it is even), but not a multiple of 3 (8 is not divisible by 3). So, 8 is not a correct answer.
(B) 9: Is a multiple of 3 (9 is divisible by 3), but not a multiple of 2 (it is odd). So, 9 is not a correct answer.
(C) 12: Is a multiple of 2 (it is even) and a multiple of 3 (1+2=3, which is divisible by 3). So, 12 is a correct answer.
(D) 18: Is a multiple of 2 (it is even) and a multiple of 3 (1+8=9, which is divisible by 3). So, 18 is a correct answer.
(E) 21: Is a multiple of 3 (2+1=3, which is divisible by 3), but not a multiple of 2 (it is odd). So, 21 is not a correct answer.
(F) 36: Is a multiple of 2 (it is even) and a multiple of 3 (3+6=9, which is divisible by 3). So, 36 is a correct answer.
Step 4: Final Answer:
The integers that are multiples of both 2 and 3 are 12, 18, and 36.