The difference between any number of four digits and the number formed by using the digits in the reversed order is exactly divisible by :

Options

A. 11
B. 10
C. 9
D. 5

Answer

To solve the problem of finding out by which number the difference between any four-digit number and the number formed by reversing its digits is exactly divisible, we can follow these steps:

### Step-by-Step Solution:

1. **Define a Four-Digit Number**:
Let the four-digit number be represented as \( abcd \), where \( a, b, c, d \) are the digits of the number. The value of this number can be expressed as:
\[ N = 1000a + 100b + 10c + d \]

2. **Reverse the Digits**:
The number formed by reversing the digits will be \( dcba \). The value of this reversed number can be expressed as:
\[ R = 1000d + 100c + 10b + a \]

3. **Calculate the Difference**:
Now, we need to find the difference between the original number \( N \) and the reversed number \( R \):
\[ \text{Difference} = N - R = (1000a + 100b + 10c + d) - (1000d + 100c + 10b + a) \]

4. **Simplify the Expression**:
Simplifying the difference gives:
\[ \text{Difference} = (1000a - a) + (100b - 10b) + (10c - 100c) + (d - 1000d) \]
\[ = 999a + 90b - 90c - 999d \]
\[ = 999(a - d) + 90(b - c) \]

5. **Factor Out Common Terms**:
Notice that both terms in the expression \( 999(a - d) + 90(b - c) \) can be factored:
\[ = 9(111(a - d) + 10(b - c)) \]

6. **Conclusion**:
Since the expression \( 9(111(a - d) + 10(b - c)) \) is clearly divisible by 9, we conclude that the difference between any four-digit number and the number formed by reversing its digits is exactly divisible by 9.

### Final Answer:
The difference between any four-digit number and the number formed by reversing its digits is exactly divisible by **9**.