If 5432*7 is divisible by 9, then the digit in place of * is

Options

A. 0
B. 1
C. 6
D. 9

Answer

To solve the problem, we need to find the digit represented by '*' in the number 5432*7 such that the entire number is divisible by 9. We will use the divisibility rule for 9, which states that a number is divisible by 9 if the sum of its digits is divisible by 9.

### Step-by-Step Solution:

1. **Identify the digits of the number**: The number is 5432*7, where '*' is the digit we need to find. We can denote '*' as 'x'. Thus, the digits are 5, 4, 3, 2, x, and 7.

2. **Calculate the sum of the digits**: We need to find the sum of these digits:
\[ 5 + 4 + 3 + 2 + x + 7 \]

3. **Perform the addition**:
- First, add 5 and 4:
\[ 5 + 4 = 9 \]
- Next, add 9 and 3:
\[ 9 + 3 = 12 \]
- Then, add 12 and 2:
\[ 12 + 2 = 14 \]
- Finally, add 14 and 7:
\[ 14 + 7 = 21 \]
- Therefore, the sum of the digits is:
\[ 21 + x \]

4. **Set up the divisibility condition**: For the number to be divisible by 9, the sum \(21 + x\) must be divisible by 9.

5. **Check possible values for x (0 to 9)**:
- We will check each digit from 0 to 9 to see which makes \(21 + x\) divisible by 9.
- Calculate \(21 + x\) for each value of x:
- If \(x = 0\): \(21 + 0 = 21\) (not divisible by 9)
- If \(x = 1\): \(21 + 1 = 22\) (not divisible by 9)
- If \(x = 2\): \(21 + 2 = 23\) (not divisible by 9)
- If \(x = 3\): \(21 + 3 = 24\) (not divisible by 9)
- If \(x = 4\): \(21 + 4 = 25\) (not divisible by 9)
- If \(x = 5\): \(21 + 5 = 26\) (not divisible by 9)
- If \(x = 6\): \(21 + 6 = 27\) (divisible by 9)
- If \(x = 7\): \(21 + 7 = 28\) (not divisible by 9)
- If \(x = 8\): \(21 + 8 = 29\) (not divisible by 9)
- If \(x = 9\): \(21 + 9 = 30\) (not divisible by 9)

6. **Conclusion**: The only value of x that makes \(21 + x\) divisible by 9 is \(x = 6\).

Thus, the digit in place of '*' is **6**.