Question

Is n an even number?
Statement I: n is divisible by 4.
Statement II: n is divisible by 8.

• A. Statement I alone is sufficient
• B. Statement II alone is sufficient
• C. Either statement alone is sufficient
• D. Both statements together are required

Hint:

If a statement allows you to answer the question with a definite "Yes" or a definite "No", it is considered sufficient.
A common mistake is thinking a statement is insufficient if it yields a "No" answer; as long as the answer is unique and definite, the data is sufficient.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This is a Data Sufficiency question concerning the properties of integers (Number Theory).
We need to determine if the integer $n$ is an even number.
An even number is any integer that is completely divisible by 2.
We need to evaluate whether Statement I alone, Statement II alone, or both are sufficient to answer this question with a definite "Yes" or "No".

Step 2: Key Formula or Approach:
An integer $n$ is even if it can be written in the form $n = 2k$, where $k$ is an integer.
If a number is divisible by any multiple of 2 (such as 4, 6, 8, etc.), it must also be divisible by 2 itself.

Step 3: Detailed Explanation:
$\bullet$ Let us evaluate Statement I alone:
$\bullet$ Statement I says: $n$ is divisible by 4.
$\bullet$ If $n$ is divisible by 4, we can write $n = 4k$, where $k$ is an integer.
$\bullet$ We can rewrite this as:
\[ n = 2 \times (2k) \]
$\bullet$ Let $m = 2k$ (which is also an integer). Then $n = 2m$.
$\bullet$ This proves that $n$ is divisible by 2, meaning $n$ is definitely an even number.
$\bullet$ Thus, Statement I alone is sufficient to answer the question with a definite "Yes".
$\bullet$ Now, let us evaluate Statement II alone:
$\bullet$ Statement II says: $n$ is divisible by 8.
$\bullet$ If $n$ is divisible by 8, we can write $n = 8j$, where $j$ is an integer.
$\bullet$ We can rewrite this as:
\[ n = 2 \times (4j) \]
$\bullet$ Let $p = 4j$ (which is an integer). Then $n = 2p$.
$\bullet$ This proves that $n$ is divisible by 2, meaning $n$ is definitely an even number.
$\bullet$ Thus, Statement II alone is also sufficient to answer the question with a definite "Yes".
$\bullet$ Since each statement alone is independently sufficient to answer the question, the correct choice is that either statement alone is sufficient.

Step 4: Final Answer:
Either statement alone is sufficient to determine that $n$ is an even number.
Therefore, the correct option is (C).