7X2 is a three digit number and X is the missing digit. If the number is divisible by 6, the missing digit is

Options

A. 5
B. 3
C. 2
D. 4

Answer

To solve the problem, we need to determine the missing digit \( X \) in the three-digit number \( 7X2 \) such that the entire number is divisible by 6. A number is divisible by 6 if it is divisible by both 2 and 3.

### Step-by-step Solution:

1. **Check Divisibility by 2:**
- The last digit of the number \( 7X2 \) is 2. Since 2 is an even number, \( 7X2 \) is divisible by 2.

2. **Check Divisibility by 3:**
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- The digits of the number \( 7X2 \) are 7, \( X \), and 2.
- We calculate the sum of the digits: \( 7 + X + 2 = 9 + X \).

3. **Set Up the Condition for Divisibility by 3:**
- We need \( 9 + X \) to be divisible by 3.
- Since 9 is already divisible by 3, \( X \) must also be such that \( X \) is divisible by 3.

4. **Identify Possible Values for \( X \):**
- The possible digits for \( X \) (which must be a single digit) are 0 through 9.
- The digits that are divisible by 3 within this range are: 0, 3, 6, and 9.

5. **Check Each Possible Value of \( X \):**
- **If \( X = 0 \):**
- \( 9 + 0 = 9 \) (divisible by 3)
- **If \( X = 3 \):**
- \( 9 + 3 = 12 \) (divisible by 3)
- **If \( X = 6 \):**
- \( 9 + 6 = 15 \) (divisible by 3)
- **If \( X = 9 \):**
- \( 9 + 9 = 18 \) (divisible by 3)

6. **Conclusion:**
- The possible values for \( X \) that make \( 7X2 \) divisible by 6 are 0, 3, 6, and 9.
- Therefore, \( X \) can be 0, 3, 6, or 9.

### Final Answer:
The missing digit \( X \) can be 0, 3, 6, or 9.