Which of the follwoing number will also divide a 6 digit number which is in the sequence of xyxyxy. (Where `1 le x le 9, 1 le y le 9)`.

Options

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

Answer

To solve the problem of finding which number can divide a six-digit number in the sequence of xyxyxy, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Number Format**:
The six-digit number in the form of xyxyxy can be expressed as:
\[ xyxyxy = xy \times 1000 + xy \times 100 + xy \times 1 \]
Here, \(xy\) represents a two-digit number formed by digits \(x\) and \(y\).

2. **Factoring the Expression**:
We can factor out \(xy\) from the expression:
\[ xyxyxy = xy \times (1000 + 100 + 1) \]
Simplifying the expression inside the parentheses:
\[ 1000 + 100 + 1 = 1101 \]
Therefore, we can rewrite the six-digit number as:
\[ xyxyxy = xy \times 1101 \]

3. **Identifying the Divisor**:
Since \(xyxyxy\) can be expressed as \(xy \times 1101\), any number that divides \(xyxyxy\) must also divide \(1101\).

4. **Finding the Factors of 1101**:
Next, we need to find the factors of \(1101\) to determine which of the given options can divide \(xyxyxy\).
To factor \(1101\):
- Check divisibility by \(3\): The sum of the digits \(1 + 1 + 0 + 1 = 3\) is divisible by \(3\).
- Dividing \(1101\) by \(3\):
\[ 1101 \div 3 = 367 \]
- Now, we check if \(367\) is a prime number. It is not divisible by \(2\), \(3\), \(5\), \(7\), \(11\), or \(13\) (up to the square root of \(367\)). Thus, \(367\) is prime.

5. **Conclusion**:
The factors of \(1101\) are \(1\), \(3\), \(367\), and \(1101\). Therefore, any number that is a factor of \(1101\) will also divide the six-digit number \(xyxyxy\).