Is the expression X − 5 an even integer, and which of the given statements about X is or are sufficient to answer this question?

Introduction / Context:
This is a data sufficiency question involving parity (even or odd) of an expression. The main task is not to find the value of X, but to decide which statement gives enough information to determine whether X − 5 is an even integer.

Given Data / Assumptions:

• Main question: Is X − 5 even?
• X is a real number.
• Statement 1: X − 15 is an integer.
• Statement 2: X − 10 is an odd integer.
• Even and odd are considered only for integers.

Concept / Approach:
To decide whether X − 5 is even, we must first be sure it is an integer and then know its parity. We examine each statement to see whether it fixes X in a way that allows us to deduce the parity of X − 5. If multiple integer values with different parity for X − 5 are possible under a statement, that statement is not sufficient.

Step-by-Step Solution:
Step 1: From statement 1, X − 15 is an integer. Let X − 15 = k, where k is any integer. Then X = k + 15 and X − 5 = k + 10. Step 2: Because k can be any integer, k + 10 can be even or odd depending on k. So we cannot decide the parity of X − 5. Statement 1 alone is insufficient. Step 3: From statement 2, X − 10 is an odd integer. Let X − 10 = 2n + 1, where n is an integer. Then X = 2n + 11. Step 4: Now X − 5 = (2n + 11) − 5 = 2n + 6 = 2(n + 3), which is clearly an even integer. Step 5: Under statement 2, X − 5 is always even, so statement 2 alone is sufficient.

Verification / Alternative check:
Choose a sample value under statement 2. If X − 10 = 3, an odd integer, then X = 13 and X − 5 = 8, which is even. If X − 10 = 5, then X = 15 and X − 5 = 10, again even. Any odd integer used in place of X − 10 will always lead to an even value for X − 5.

Why Other Options Are Wrong:
Option A wrongly claims that statement 1 is sufficient, even though it leaves parity undecided. Option B says neither is sufficient, but we have seen statement 2 alone is enough. Option D claims both are needed, but statement 2 alone already answers the question. Option E suggests that the combination is required, which overcomplicates the situation.

Common Pitfalls:
One common mistake is to ignore the requirement that even and odd apply only to integers and to overlook that statement 1 allows many different integer values. Another error is failing to express X in algebraic form from each statement, which makes parity reasoning less clear.

Final Answer:
The correct conclusion is that statement 2 alone is sufficient while statement 1 alone is insufficient to determine whether X − 5 is even.