If a four digit number is formed by repeating a two digit number two times (e.g. 2525), then that number will be divisible by :

Options

A. 1001
B. 101
C. 10001
D. 10101

Answer

To solve the problem, we need to understand how a four-digit number formed by repeating a two-digit number can be expressed mathematically. Let's break it down step by step.

### Step 1: Represent the two-digit number
Let the two-digit number be represented as \( xy \), where \( x \) and \( y \) are the digits of the number.

### Step 2: Form the four-digit number
When we repeat this two-digit number twice, we form a four-digit number. This can be represented as:
\[ N = xyxy \]
This means \( N \) can be expressed as:
\[ N = 1000x + 100y + 10x + y = 1010x + 101y \]

### Step 3: Factor the expression
Now, we can factor out the common term from the expression:
\[ N = 101(10x + y) \]
Here, \( 10x + y \) is simply the original two-digit number \( xy \).

### Step 4: Analyze divisibility
From the expression \( N = 101(10x + y) \), we can see that \( N \) is clearly divisible by 101, since it is a product of 101 and another integer \( (10x + y) \).

### Conclusion
Thus, any four-digit number formed by repeating a two-digit number will always be divisible by 101.