Given that a and b are integers and a + a^2*b^3 is odd, which of the following statements is correct? A. a and b are both odd B. a and b are both even C. a is even and b is odd D. a is odd and b is even

GATE 2018 · General Aptitude · Quadratic Equations · medium

Answer: a is odd and b is even, i.e., option D.

1. Factor the expression: Factor a: a + a^2*b^3 = a(1 + a*b^3). For this product to be odd, both a and (1 + a*b^3) must be odd.
2. Determine parity of a: Since a*(1 + a*b^3) is odd, a must be odd. (If a were even, the entire product would be even.)
3. Determine parity of b: With a odd, 1 + a*b^3 is odd iff a*b^3 is even iff b^3 is even iff b is even. So b must be even. Option A (both odd) is eliminated since b must be even. Option D (a odd, b even) is the answer.