Correct Option (D)
A six-digit number of the form XYZXYZ can be expressed algebraically by expanding its place values. If XYZ represents a three-digit number, then the six-digit number XYZXYZ can be written as:
XYZXYZ = 1000 × (XYZ) + (XYZ)
= (1000 + 1) × (XYZ)
= 1001 × (XYZ)
To determine the divisors of any number of the form XYZXYZ, the prime factors of 1001 must be identified. The prime factorization of 1001 is 7 × 11 × 13.
Therefore, XYZXYZ = 7 × 11 × 13 × (XYZ).
This algebraic expansion demonstrates that any number of the form XYZXYZ is inherently divisible by 7, 11, and 13, irrespective of the specific integer values chosen for X, Y, and Z (provided XYZ forms a valid three-digit number).
Incorrect Options:
Options (A), (B), and (C) are incorrect because they propose divisibility by only a subset of the prime factors (7, 11, 13) or an incomplete combination. As established, the number XYZXYZ is demonstrably divisible by all three prime numbers (7, 11, and 13). Therefore, any option suggesting divisibility by only two of these factors is incomplete and does not fully represent the divisibility properties of the number.