Question

What is the value of x?
(1) \((x)(x + 1) = (2013)(2014)\)
(2) x is odd

• A. Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
• B. EACH statement ALONE is sufficient to answer the question asked
• C. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
• D. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• E. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

Hint:

When you see a quadratic equation in a data sufficiency problem, always look for both solutions. A common trap is to find one obvious solution by inspection and forget to look for the other, especially if it's negative.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a "value" data sufficiency question. We need to determine if the given statements, alone or together, can narrow down the possibilities for x to a single, unique value.

Step 2: Key Formula or Approach:
Statement (1) provides a quadratic equation. We need to find all possible solutions for x. Statement (2) provides a property of x. We will then combine the information to see if a unique solution emerges.

Step 3: Detailed Explanation:
Analyzing Statement (1): \((x)(x + 1) = (2013)(2014)\).
This equation is in the form of a product of two consecutive numbers. By simple inspection, one obvious solution is \(x = 2013\), because if \(x=2013\), then \(x+1=2014\), and the equation becomes \((2013)(2014) = (2013)(2014)\), which is true.
However, since this is a quadratic equation (\(x^2 + x - (2013)(2014) = 0\)), there might be another solution. Let's consider the case with negative numbers. Let \(y = -x\). Then the equation is \((-y)(-y+1) = (2013)(2014)\), which simplifies to \(y(y-1) = (2013)(2014)\). Alternatively, let's look at the structure \(x(x+1)\). The product of two consecutive negative integers is positive. Let \(x+1 = -2013\). Then \(x = -2014\). Let's check this solution: If \(x = -2014\), then \(x+1 = -2013\). The product is \((-2014)(-2013) = (2014)(2013)\), which is also true. So, from statement (1), x could be 2013 or -2014. Since there are two possible values, statement (1) alone is not sufficient.
Analyzing Statement (2): x is odd.
This statement tells us that x belongs to the set \(\{..., -3, -1, 1, 3, ...\}\). This information on its own is clearly not enough to find a unique value for x. So, statement (2) alone is not sufficient.
Analyzing Both Statements Together:
From statement (1), we found that \(x = 2013\) or \(x = -2014\). From statement (2), we know that x must be an odd number. Let's check our two possible values: - Is 2013 odd? Yes, its last digit is 3. - Is -2014 odd? No, it is an even number. By combining both statements, we eliminate \(x = -2014\), leaving only one possible value: \(x = 2013\). Since we have found a unique value for x, both statements together are sufficient.

Step 4: Final Answer:
Neither statement is sufficient on its own, but together they provide enough information to determine a unique value for x.