If a six digit number is formed by repeating a three digit number (e.g. 656656, 214214), then that number will be divisible by :

Options

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

Answer

To solve the problem of determining the divisibility of a six-digit number formed by repeating a three-digit number, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure of the Number**:
A six-digit number formed by repeating a three-digit number can be represented as \( ABCABC \), where \( ABC \) is a three-digit number.

2. **Express the Number Mathematically**:
The number \( ABCABC \) can be expressed as:
\[ ABCABC = ABC \times 1000 + ABC \]
This simplifies to:
\[ ABCABC = ABC \times (1000 + 1) = ABC \times 1001 \]

3. **Identify the Factors of 1001**:
Now, we need to find out what numbers \( ABC \) must be divisible by. Since \( ABCABC \) is \( ABC \times 1001 \), it is clear that \( ABCABC \) will be divisible by \( 1001 \).

4. **Factorization of 1001**:
To understand the divisibility better, we can factor \( 1001 \):
\[ 1001 = 7 \times 11 \times 13 \]
Therefore, any number that is a multiple of \( 1001 \) is also divisible by \( 7 \), \( 11 \), and \( 13 \).

5. **Conclusion**:
Since \( ABCABC \) is divisible by \( 1001 \), it follows that it is also divisible by \( 7 \), \( 11 \), and \( 13 \). Thus, the six-digit number formed by repeating a three-digit number will be divisible by \( 1001 \).

### Final Answer:
The number will be divisible by \( 1001 \), which is \( 7 \), \( 11 \), and \( 13 \).

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