If x is divisible by both 3 and 4, then the number x must be a multiple of which of the following?

Options

A. 8
B. 12
C. 15
D. 18

Answer

To solve the problem, we need to determine what number \( x \) must be a multiple of if it is divisible by both 3 and 4.

### Step-by-Step Solution:

1. **Understanding Divisibility**:
We start by noting that if a number \( x \) is divisible by both 3 and 4, it means that \( x \) can be expressed as:
\[ x = 3k \quad \text{for some integer } k \]
and
\[ x = 4m \quad \text{for some integer } m. \]

2. **Finding the LCM**:
To find a common multiple of 3 and 4, we need to calculate the Least Common Multiple (LCM) of these two numbers. The LCM is the smallest number that is a multiple of both.

- The prime factorization of 3 is \( 3^1 \).
- The prime factorization of 4 is \( 2^2 \).

The LCM is found by taking the highest power of each prime that appears in the factorizations:
\[ \text{LCM}(3, 4) = 2^2 \times 3^1 = 4 \times 3 = 12. \]

3. **Conclusion**:
Since \( x \) is divisible by both 3 and 4, it must also be a multiple of their LCM, which is 12. Therefore, we conclude that:
\[ x \text{ must be a multiple of } 12. \]

4. **Identifying the Correct Option**:
Among the given options:
- Option A: 8
- Option B: 12 (Correct)
- Option C: 15
- Option D: 18

The correct answer is option B, which is 12.

### Final Answer:
Thus, if \( x \) is divisible by both 3 and 4, then \( x \) must be a multiple of **12**.

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