Which of the following numbers will always divide a six-digit number of the form xyxyxy `("where" 1 le x le 9, le y le 9)` ?

Options

A. 1010
B. 10101
C. 11011
D. 11010

Answer

To determine which number will always divide a six-digit number of the form \(xyxyxy\) (where \(1 \leq x \leq 9\) and \(0 \leq y \leq 9\)), we can break down the number and analyze its structure.

### Step-by-Step Solution:

1. **Understanding the Structure of the Number**:
The six-digit number \(xyxyxy\) can be expressed in terms of its place values:
\[ xyxyxy = 100000x + 10000y + 1000x + 100y + 10x + y \]

2. **Combining Like Terms**:
We can combine the terms involving \(x\) and \(y\):
\[ = (100000x + 1000x + 10x) + (10000y + 100y + y) \]
\[ = (100000 + 1000 + 10)x + (10000 + 100 + 1)y \]
\[ = 101010x + 10101y \]

3. **Factoring the Expression**:
We can factor out the common term:
\[ = 10101(10x + y) \]

4. **Conclusion**:
From the expression \(10101(10x + y)\), we can see that \(10101\) is a constant factor of the number \(xyxyxy\). Therefore, any six-digit number of the form \(xyxyxy\) will always be divisible by \(10101\).

### Final Answer:
The number that will always divide a six-digit number of the form \(xyxyxy\) is **10101**.