Question

If \(m,n\) are two positive integers, is \(m\) an odd integer? Statements: (I) \(m+n\) is an odd integer. (II) \(mn\) is an even integer.

• A. Statement (I) alone is sufficient to answer the question, but (II) alone is not sufficient.
• B. Statement (II) alone is sufficient to answer the question, but (I) alone is not sufficient.
• C. Both the statements (I) and (II) are sufficient to answer the question, but neither statement alone is sufficient.
• D. Both the statements (I) and (II) together are not sufficient to answer the question and additional data are required.

Hint:

For parity-based data sufficiency questions, always test multiple valid examples. If the answer changes, the data is not sufficient.

Answer 1

The Correct Option is D

Concept: To determine whether \(m\) is odd, we use the properties of odd and even numbers:
• Odd + Even = Odd
• Odd + Odd = Even
• Even + Even = Even
• Even \(\times\) Any number = Even
• Odd \(\times\) Odd = Odd In data sufficiency, the statements are sufficient only if they give a unique answer.

Step 1: Checking Statement (I): \(m+n\) is odd.
If the sum of two integers is odd, one must be odd and the other even. Possible cases: \[ m=3,\; n=2 \] Here, \(m\) is odd. Another case: \[ m=2,\; n=3 \] Here, \(m\) is even. Thus, \(m\) can be odd or even. So, Statement (I) alone is not sufficient.

Step 2: Checking Statement (II): \(mn\) is even.
If the product is even, at least one of the numbers must be even. Possible cases: \[ m=2,\; n=3 \] Here, \(m\) is even. Another case: \[ m=3,\; n=2 \] Here, \(m\) is odd. Thus, \(m\) can again be odd or even. So, Statement (II) alone is not sufficient.

Step 3: Checking both statements together.
From Statement (I): \[ m+n=\text{odd} \] So one number is odd and the other is even. From Statement (II): \[ mn=\text{even} \] This only confirms that one number is even. Still, we cannot determine whether \(m\) specifically is odd. Case 1: \[ m=3,\; n=2 \] Here \(m\) is odd. Case 2: \[ m=2,\; n=3 \] Here \(m\) is even. Since both cases satisfy both statements but give different answers, the data is insufficient. Hence, additional information is required.