Points to remember:

•   Divisibility by 7: If we subtract twice the last digit from the rest of the number, and the result is divisible by 7, then the original number is divisible by 7.

•   Divisibility by 11: If the difference between the sum of the digits at odd places and the sum of the digits at even places is divisible by 11 (or zero), then the original number is divisible by 11.

•   Divisibility by 13: To check if a number is divisible by 13, multiply its unit place digit by 4, then add the product obtained to the number formed by the rest of the digits of the number. If the result is 0 or a multiple of 13, the result is divisible by 13.

Explanation:

XYZXYZ = XYZ000 + XYZ

= XYZ (1000 + 1)

= XYZ × 1001

•   The number 1001 can be factored into 7, 11, and 13.

•   Therefore, any number in the form XYZXYZ must be divisible by 7, 11, and 13.

•   Thus, option (d) is the correct answer.

Exercise:

Checking if 1001 is divisible by 13.

•   Multiply its unit place digit by 4 => 1*4 = 4

•   Add the product obtained to the number formed by the rest of the digits of the number => 100+4 = 104

•   Check if 104 is 0 or a multiple of 13, 1001 will be divisible by 13. (13*8 = 104)

•   Hence, 1001 is divisible by 13.