Question

If m is an integer, is m odd?
(1) \(1 + m^2\) is an odd integer.
(2) \(5m - 2\) is an even integer.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

For Yes/No data sufficiency questions, a statement is sufficient if it gives a conclusive answer, either always "Yes" or always "No". If a statement allows for both possibilities, it is not sufficient.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question is a Yes/No question about the parity (evenness or oddness) of an integer \(m\). A statement is sufficient if it allows us to give a definitive "Yes" or a definitive "No" answer. We need to use the properties of even and odd numbers.
- Even + Even = Even; Odd + Odd = Even; Even + Odd = Odd
- Even \(\times\) Any Integer = Even; Odd \(\times\) Odd = Odd
Step 2: Detailed Explanation:
Analyzing Statement (1):
We are given that \(1 + m^2\) is an odd integer.
Since 1 is an odd number, we have:
\[ \text{Odd} + m^2 = \text{Odd} \]
For this to be true, \(m^2\) must be an even number (because Odd + Even = Odd).
If \(m^2\) is even, then the integer \(m\) must also be even. (If \(m\) were odd, \(m^2 = m \times m = \text{odd} \times \text{odd} = \text{odd}\)).
So, statement (1) implies that \(m\) is even.
This provides a definitive "No" to the question "is m odd?". Therefore, statement (1) alone is sufficient.
Analyzing Statement (2):
We are given that \(5m - 2\) is an even integer.
Since 2 is an even number, we have:
\[ 5m - \text{Even} = \text{Even} \]
For this to be true, \(5m\) must be an even number (because Even - Even = Even).
The product of two integers is even if at least one of them is even. Since 5 is an odd number, for the product \(5m\) to be even, \(m\) must be an even integer.
So, statement (2) implies that \(m\) is even.
This also provides a definitive "No" to the question "is m odd?". Therefore, statement (2) alone is sufficient.
Step 3: Final Answer:
Both statements independently lead to the conclusion that \(m\) is even, thus providing a definite "No" to the question. Therefore, each statement alone is sufficient.