A six digit number is formed by repeating a three digit number, for example, 256256 or 678678 etc. Any number of this form is always exactly divisible by :

Options

A. 7 only
B. 11 only
C. 13 only
D. 1001

Answer

To solve the problem, we need to analyze the structure of a six-digit number formed by repeating a three-digit number. Let's denote the three-digit number as \( ABC \). The six-digit number can be expressed as \( ABCABC \).

### Step-by-Step Solution:

1. **Understanding the Structure**:
- A six-digit number formed by repeating a three-digit number can be written as \( ABCABC \).
- For example, if \( ABC = 256 \), then \( ABCABC = 256256 \).

2. **Mathematical Representation**:
- We can express \( ABCABC \) mathematically:
\[ ABCABC = ABC \times 1000 + ABC \]
- This is because \( ABC \) is shifted three places to the left (multiplied by 1000) and then added to itself.

3. **Factoring the Expression**:
- We can factor out \( ABC \):
\[ ABCABC = ABC \times (1000 + 1) = ABC \times 1001 \]

4. **Divisibility**:
- From the expression \( ABC \times 1001 \), we can conclude that \( ABCABC \) is divisible by both \( ABC \) and \( 1001 \).
- Since \( ABC \) is a three-digit number, it does not affect the divisibility by \( 1001 \).

5. **Conclusion**:
- Therefore, any six-digit number formed by repeating a three-digit number is always exactly divisible by \( 1001 \).

### Final Answer:
The six-digit number formed by repeating a three-digit number is exactly divisible by **1001**.