A four digit number is formed repeating two digits two times. Like 2525, 3232 etc. such type of number always divisible by?

Options

A. 7
B. 101
C. 12
D. 3

Answer

To solve the problem of determining what a four-digit number formed by repeating two digits twice (like 2525 or 3232) is always divisible by, we can follow these steps:

### Step 1: Understand the Structure of the Number
A four-digit number formed by repeating two digits can be represented as \(ABAB\), where \(A\) and \(B\) are the two digits.

### Step 2: Express the Number Mathematically
The number \(ABAB\) can be expressed in terms of its digits:
\[ ABAB = 1000A + 100B + 10A + B = 1010A + 101B \]

### Step 3: Factor the Expression
We can factor out common terms from the expression:
\[ ABAB = 101(10A + B) \]
This shows that \(ABAB\) is equal to \(101\) multiplied by another integer \((10A + B)\).

### Step 4: Conclusion on Divisibility
Since \(ABAB\) can be expressed as \(101\) times another integer, it follows that \(ABAB\) is always divisible by \(101\).

### Final Answer
Thus, any four-digit number formed by repeating two digits twice is always divisible by **101**.

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