If A = {x : x is a multiple of 4} and B = {x : x is a multiple of 6}, then A `sub` B consists of all multiple of

Options

A. 4
B. 8
C. 12
D. 16

Answer

To solve the problem, we need to determine the relationship between the sets A and B, specifically focusing on the multiples of 4 and 6.

### Step-by-Step Solution:

1. **Define Set A**:
- Set A consists of all multiples of 4.
- Therefore, A = {4, 8, 12, 16, 20, 24, ...}.

2. **Define Set B**:
- Set B consists of all multiples of 6.
- Therefore, B = {6, 12, 18, 24, 30, ...}.

3. **Identify the Intersection of A and B**:
- We need to find the common elements in both sets A and B.
- The common multiples of 4 and 6 are the multiples of the least common multiple (LCM) of 4 and 6.

4. **Calculate the LCM of 4 and 6**:
- The prime factorization of 4 is 2², and for 6 it is 2¹ * 3¹.
- The LCM is found by taking the highest power of each prime factor:
- LCM(4, 6) = 2² * 3¹ = 12.

5. **Conclusion**:
- Therefore, the intersection of sets A and B, denoted as A ∩ B, consists of all multiples of 12.
- Hence, A `subset` B consists of all multiples of 12.

### Final Answer:
A `subset` B consists of all multiples of 12.

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