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

(a) 7 and 11 only

(b) 11 and 13 only

(c) 7 and 13 only

(d) 7, 11 and 13

Correct Answer: (d) 7, 11 and 13

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

## Number XYZXYZ Analysis

The number of the form XYZXYZ can be expressed as follows:

XYZXYZ = XYZ * 1000 + XYZ

Now, factor out 1000 from the first term:

XYZXYZ = XYZ * (1000 + 1)

Simplify the expression inside the parentheses:

XYZXYZ = XYZ * 1001

So, the number of the form XYZXYZ is equal to XYZ * 1001.

The prime factorization of 1001 is 7 * 11 * 13. Therefore, any number of the form XYZ * 1001 will be divisible by 7, 11, and 13.

So, the correct answer is (d) 7, 11, and 13. You can verify this by dividing 111111 by 7, 11, or 13 and getting a whole number as the quotient.