For a positive integer n, the function @n represents the product of al

Ans is D .

@n = product of even numbers between n and 2n , exclusive n and 2n , and must contain all even numbers.

I. 9

-- @9 = 10 * 12 ......*16 [Max prime factor is 7 . so k = 9 not possible ]


II. 18

@18 = 20 * 22 ......*34 [Max prime factor is 17 . so k = 18 possible ]


III. 32
-- @32 = 34 * 12 ......*64 [Max prime factor is 31 . so k = 32 not possible ]

Hence ans is D

The question is which cannot be the value of k

From your solution k can be 18 but cannot be 9 or 32

So the answer is D

­I. If k=9 then the range is 9 to 18 -> 17 cannot be a factor of the product. Cancelled.
II. If k=18 then the range is 18 to 36 -> the maximum even integer in the range is 34=2*17. Keep this.
III. If k=32 then the range is 32 to 64 -> the maximum even integer in the range is 62=2*31. Cancelled.

I and III cannot be the value of k. Option (D) is correct.

My explanation with a tad bit more detail:

I) @9 = 10 * 12 * 14 * 16 (because 2n = 2 * 9 = 18)-> the prime factor of this product is not 17 at all, because none of the numbers here are divisible by 17. Since 9 cannot be k, this answer choice is correct
II) @18 = 20 * 22 * 24 * 26 * 28 * 30 * 32 * 34 (because 2n = 2 * 18 = 36) -> the biggest prime factor of this product actually IS 17, because 34 is divisible by 17, and there are no other prime products of a prime number and 2 in this sequence in which the prime number is bigger than 17. Since 18 is a valid value of k, this answer choice is incorrect.
III) @32 = 34 * 36 * 38 * 40 * ... * 62 (because 2n = 2 * 32 = 64) -> while 17 is a prime factor of this product due to the value of 34, it is NOT the biggest one. That would be 31, because of 62 (31 * 2) due to the same logic explained in II. Since 32 cannot be k, this answer choice is correct.

Choice I + III, D is the correct answer.­