For any choices of values of X,Y and Z, the 6 digit number of the form XYZXYZ is divisible by :
- A7 and 11 only
- B11 and 13 only
- C7 and 13 only
- D7, 11 and 13Correct
Explanation
The 6-digit number XYZXYZ can be written as XYZ * 1000 + XYZ. This simplifies to XYZ * (1000 + 1) = XYZ * 1001.
Now, we need to find the prime factors of 1001. 1001 divided by 7 is 143. 143 divided by 11 is 13. So, 1001 = 7 * 11 * 13.
Therefore, the number XYZXYZ = XYZ * 7 * 11 * 13. This shows that the number XYZXYZ is always divisible by 7, 11, and 13, regardless of the values of X, Y, and Z.
Analyzing the options: A) 7 and 11 only - Incorrect, it is also divisible by 13. B) 11 and 13 only - Incorrect, it is also divisible by 7. C) 7 and 13 only - Incorrect, it is also divisible by 11. D) 7, 11 and 13 - Correct, as the number is a product of XYZ and 7, 11, and 13.
The final answer is D.
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