For any choices of values of X, Y and Z, the 6 digit number of the form XYZXYZ is divisible by:

1. A. 7 and 11 only
2. B. 11 and 13 only
3. C. 7 and 13 only
4. D. 7, 11 and 13

Answer: D

Explanation

A 6-digit number of the form XYZXYZ can be expressed mathematically as:
XYZXYZ = XYZ × 1000 + XYZ
= XYZ × (1000 + 1)
= XYZ × 1001
Now, we need to find the prime factors of 1001.
1001 can be divided by 7: 1001 ÷ 7 = 143.
143 can be divided by 11: 143 ÷ 11 = 13.
13 is a prime number.
So, the prime factorization of 1001 is 7 × 11 × 13.
Therefore, any number of the form XYZXYZ is always divisible by 7, 11, and 13.