Understanding the Question: Is (X – 5) Even?

The question asks whether the expression &((X – 5)&) is an even number. For an expression to be an even number, it must satisfy two conditions:

1. It must be an integer.
2. That integer must be divisible by 2.

We are given that &((X)&) is a real number. We need to evaluate if the provided statements give us enough information to definitively answer whether &((X – 5)&) is even (yes or no).

Analyzing Statement 1: X – 15 belongs to integer

Statement 1 tells us that &((X – 15)&) is an integer. Let's represent this integer by &$(k)&$. So, &((X – 15 = k)&), where &$(k)&) is an integer.

We want to know if &((X – 5)&) is even. Let's try to express &((X – 5)&) in terms of &((X – 15)&):

\(X – 5 = (X – 15) + 10\)

Substitute &((X – 15 = k)&) into the expression:

\(X – 5 = k + 10\)

Since &$(k)&) is an integer, &((k + 10)&) must also be an integer. This satisfies the first condition for &((X – 5)&) to be an even number (it is an integer).

Now let's check the second condition: Is &((k + 10)&) always divisible by 2 (i.e., is it always even)?

• If &$(k)&) is an even integer (e.g., 2), then &((k + 10)&) is &((2 + 10 = 12)&), which is even.
• If &$(k)&) is an odd integer (e.g., 3), then &((k + 10)&) is &((3 + 10 = 13)&), which is odd.

The parity of &((X – 5)&) depends on the parity of &$(k)&), which is &((X – 15)&). Statement 1 only says &((X – 15)&) is an integer; it does not specify if it's even or odd.

Therefore, Statement 1 alone is not sufficient to determine whether &((X – 5)&) is even.

Analyzing Statement 2: X – 10 is an odd integer

Statement 2 tells us that &((X – 10)&) is an odd integer. Let's represent this odd integer by &$(m)&$. So, &((X – 10 = m)&), where &$(m)&) is an odd integer.

An odd integer can be written in the form &((2j + 1)&) for some integer &$(j)&)$. So, &((m = 2j + 1)&).

We want to know if &((X – 5)&) is even. Let's try to express &((X – 5)&) in terms of &((X – 10)&):

\(X – 5 = (X – 10) + 5\)

Substitute &((X – 10 = m)&) into the expression:

\(X – 5 = m + 5\)

Since &$(m)&) is an odd integer, let's substitute &((m = 2j + 1)&):

\(X – 5 = (2j + 1) + 5\)

\(X – 5 = 2j + 6\)

\(X – 5 = 2(j + 3)\)

Since &$(j)&) is an integer, &((j + 3)&) is also an integer. Let &((p = j + 3)&). Then &((X – 5 = 2p)&), where &$(p)&) is an integer.

By the definition of an even number, &((2p)&) is always an even integer, regardless of the specific integer value of &$(p)&)$.

Therefore, if &((X – 10)&) is an odd integer, then &((X – 5)&) is always an even integer. Statement 2 alone is sufficient to definitively answer that &((X – 5)&) is even.

Conclusion: Evaluating Sufficiency of Statements

Based on our analysis:

• Statement 1 alone is insufficient because the parity of &((X – 5)&) depends on the unknown parity of &((X – 15)&).
• Statement 2 alone is sufficient because if &((X – 10)&) is an odd integer, &((X – 5)&) is always an even integer.

Thus, Statement 2 alone provides enough information to answer the question, while Statement 1 alone does not.

Revision Table: Key Concepts Review

Additional Information: Data Sufficiency Strategy

Data Sufficiency questions require you to determine if the given statements provide enough information to answer the question, not necessarily to find the answer itself. The standard approach is:

1. Analyze the Question: Understand exactly what is being asked. What type of value are you looking for? What property needs to be satisfied (e.g., is it even, is it positive, what is its specific value)?
2. Analyze Statement 1 Alone: Assume Statement 1 is true and Statement 2 is false. Can you definitively answer the question? If yes, Statement 1 is sufficient. If no (you can find cases where the answer is "yes" and cases where it's "no"), Statement 1 is insufficient.
3. Analyze Statement 2 Alone: Assume Statement 2 is true and Statement 1 is false. Can you definitively answer the question? If yes, Statement 2 is sufficient. If no, Statement 2 is insufficient.
4. Analyze Both Statements Together: If neither Statement 1 nor Statement 2 alone is sufficient, assume both statements are true simultaneously. Can you definitively answer the question now? If yes, both are sufficient. If no, both are insufficient.

For this specific question about even/odd properties, understanding how adding or subtracting integers affects parity is crucial. Shifting an expression by an even number does not change its parity (\((n + \text{even})\) has the same parity as \(n\)). Shifting an expression by an odd number *does* change its parity (\((n + \text{odd})\) has the opposite parity as \(n\)).

In our case:

• \(X - 5 = (X - 15) + 10\): Shifting by +10 (even) means the parity of \(X - 5\) is the same as the parity of \(X - 15\). Statement 1 tells us \(X - 15\) is an integer, but not its parity.
• \(X - 5 = (X - 10) + 5\): Shifting by +5 (odd) means the parity of \(X - 5\) is the opposite of the parity of \(X - 10\). Statement 2 tells us \(X - 10\) is an odd integer. If \(X - 10\) is odd, its opposite parity is even. So, \(X - 5\) must be even.