If a and b are positive integers, is (a + b)^2 odd?

Step 1: Analyse Question Stem

a and b are positive integers; therefore, each of them are either odd or even.
We have to find out if \((a + b)^2\) is odd.

\((a + b)^2\) will be odd if (a + b) is odd. (a + b) will be odd when one of them is odd and the other is even.
Therefore, the question we are trying to answer is if a and b are of opposite natures.

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: ab is even
If ab is even, a and b could both be even; are a and b of opposite natures? NO
If ab is even, one of them could be even and the other could be odd; are a and b of opposite natures? YES

The data in statement 1 is insufficient to answer the question with a definite YES or NO.
Statement 1 alone is insufficient. Answer options A and D can be eliminated.

Statement 2: \(\frac{a}{b}\) is even

If \(\frac{a}{b}\) = even, both a and b could be even; this answers the question with a NO

If \(\frac{a}{b}\) = even, a could be even and b could be odd; this answers the question with a YES.

The data in statement 2 is insufficient to answer the question with a definite YES or NO.
Statement 2 alone is insufficient. Answer option B can be eliminated.

Step 3: Analyse Statements by combining

From statement 1: ab is even
From statement 2: \(\frac{a}{b}\) is even

As seen from the analysis of the individual statements, a and b could be of same OR opposite natures and still satisfy both the constraints.

The combination of statements is insufficient to answer the question with a definite YES or NO.
Statements 1 and 2 together are insufficient. Answer option C can be eliminated.

The correct answer option is E.