4GMAT_Mumbai's answer is a good one. I would also like to suggest a slightly different method of solving this.

To simplify: (35^2-1)/k

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Step 1) There is a shortcut to figure out the square of an integer ending with '5'.
Drop the ending '5', and multiply the (remaining number) with (remaining number + 1)
Then append '25' at the end of the result.

So:
35^2
= 3 * (3+1) <== Drop the ending '5', and multiply the (remaining number) with (remaining number + 1)
= 3*4
= 12
= 1225 <== Then append '25' at the end of the result.

Step 2)
(35^2-1)/k can be then written as:
(1225 - 1)/k
= 1224/k

Step 3)
Factorize 1224 into all it's prime factors

1224
2 * 612
2 * 2 * 306
2 * 2 * 2 * 153
2 * 2 * 2 * 3 * 51
2 * 2 * 2 * 3 * 3 * 17

Step 4)
(2 * 2 * 2 * 3 * 3 * 17) / k

Step 5) Now you can plug in the answer choices for 'k' one by one into the equation above and check which one DOES NOT result in an integer.

(A) 8 <-- 8 =2*2*2 which goes evenly in the numerator
(B) 9 <-- 9 = 3*3 which goes evenly in the numerator
(C) 12 <-- 3*2*2 which goes evenly in the numerator
(D) 16 <-- 16 = 2*2*2*2 - DOES NOT GO EVENLY IN THE NUMERATOR - This is our answer.
(E) 17 <-- goes evenly in the numerator