If k is an integer and x(x k) = k + 1, what is the value of x?

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The original condition can be modified as follows.
x(x-k) = k+1
⇔ x^2 - xk = k + 1
⇔ x^2 - xk - k - 1 = 0
⇔ x^2 -k(x+1) - 1 = 0
⇔ (x+1)(x-1) -k(x+1) = 0
⇔ (x+1)(x-k-1) = 0

Since we have 2 variable (x and y) and 1 equation, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
Since x < k, x - k - 1 ≠ 0.
Thus we have x = -1.
Condition 1) is sufficient.

Condition 2)
We have two solutions.
x = -1, k = 4
x = 2, k = 1
Thus condition 2) is not sufficient.

Therefore, the answer is A.