Understanding the Data Sufficiency Problem

The question asks for the value of 'x', which is defined as the smaller of two positive consecutive even numbers. We need to determine which of the given statements, individually or together, is sufficient to find a unique value for x.

Let the two positive consecutive even numbers be represented by x and x+2. Since they are positive, x > 0. Since they are even, x is an even integer.

Analyzing Statement I: Sum of squares of positive consecutive even numbers is 52

Statement I gives us an equation based on the property of the two numbers. Let the smaller number be x. The larger number is x+2.

According to Statement I:

\(x^2 + (x+2)^2 = 52\)

Let's expand and solve this equation:

\(x^2 + (x^2 + 4x + 4) = 52\)

\(2x^2 + 4x + 4 = 52\)

Subtract 52 from both sides:

\(2x^2 + 4x - 48 = 0\)

Divide the entire equation by 2 to simplify:

\(x^2 + 2x - 24 = 0\)

This is a quadratic equation. We can solve it by factoring. We need two numbers that multiply to -24 and add up to 2. These numbers are 6 and -4.

So, the equation can be factored as:

\((x+6)(x-4) = 0\)

This gives two possible values for x:

• \(x+6 = 0 \Rightarrow x = -6\)
• \(x-4 = 0 \Rightarrow x = 4\)

Now, we must check these values against the conditions given in the question stem. The numbers must be positive consecutive even numbers. x is the smaller number.

• If \(x = -6\), the numbers would be -6 and -4. These are not positive numbers. So, \(x = -6\) is not a valid solution according to the question's conditions.
• If \(x = 4\), the numbers would be 4 and \(4+2 = 6\). These are positive numbers, and they are consecutive even numbers. x=4 is also an even number. This solution fits all the conditions.

Since Statement I leads to a unique valid value for x (which is 4), Statement I alone is sufficient to answer the question.

Analyzing Statement II: Difference of the number is 2

Statement II says the difference between the two numbers is 2. The question stem already defines the numbers as "positive consecutive even numbers". By definition, any two consecutive even numbers (like 2 and 4, 8 and 10, 100 and 102) will always have a difference of 2.

Statement II provides information that is already implied by the question itself. It does not add any new constraint that would help us find the specific values of the two numbers. Many pairs of positive consecutive even numbers have a difference of 2 (e.g., (2,4), (4,6), (6,8), etc.). The smaller number 'x' could be 2, 4, 6, and so on.

Therefore, Statement II alone is not sufficient to find a unique value for x.

Conclusion on Sufficiency

• Statement I alone is sufficient because it uniquely determines the smaller positive consecutive even number to be 4.
• Statement II alone is not sufficient because it only provides information already known from the question stem and does not narrow down the possible values of x.

Thus, only Statement I is sufficient to answer the question.

Revision Table: Data Sufficiency Analysis

Additional Information: Consecutive Numbers and Data Sufficiency

Understanding the properties of numbers is crucial for solving quantitative problems. Consecutive even numbers are even integers that follow each other, with a difference of 2 (e.g., n, n+2, n+4...). Similarly, consecutive odd numbers have a difference of 2 (e.g., n, n+2, n+4...) where n is odd, and consecutive integers have a difference of 1 (e.g., n, n+1, n+2...).

In data sufficiency questions, we assess whether the given statements provide enough information to answer the question definitively. A statement is sufficient if it leads to a single, unique answer. If a statement leads to multiple possible answers or no answer, it is not sufficient. If combining statements leads to a unique answer when neither alone does, then both are sufficient.

In this specific data sufficiency problem, Statement I provided a unique constraint (the sum of squares) that, when combined with the initial conditions (positive, consecutive, even), pinpointed the exact numbers. Statement II simply reiterated a known property of consecutive even numbers, offering no new discriminating information.