A 4-digit number is formed by repeating a 2-digit number such as 2525, 3232, etc. Any number of this form is always exactly divisible by :

Options

A. a) 7
B. b) 11
C. c) 13
D. d) Smallest 3-digit prime number

Answer

To solve the problem of determining what a 4-digit number formed by repeating a 2-digit number is always exactly divisible by, we can follow these steps:

### Step 1: Understand the Structure of the 4-Digit Number
A 4-digit number formed by repeating a 2-digit number can be expressed in the form:
\[ \text{ABAB} = 1000A + 100B + 10A + B = 1010A + 101B \]
where A and B are the digits of the 2-digit number.

### Step 2: Factor the Expression
The expression \( 1010A + 101B \) can be factored:
\[ 1010A + 101B = 101(10A + B) \]
This shows that the 4-digit number is divisible by 101.

### Step 3: Verify with Examples
Let’s verify this with some examples:
- For the number 2525:
\[ 2525 = 101 \times 25 \]
- For the number 3232:
\[ 3232 = 101 \times 32 \]
In both cases, the 4-digit number is divisible by 101.

### Step 4: Conclusion
Since we have shown that any 4-digit number formed by repeating a 2-digit number can be expressed as \( 101(10A + B) \), we conclude that such numbers are always divisible by 101.

### Final Answer
Any 4-digit number formed by repeating a 2-digit number is always exactly divisible by **101**.

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