GATE Mechanical Engineering (ME) 2018 Shift 1 Solved Paper

Given: a and b are integers
$\mathit{a}+{\mathit{a}}^2{\mathit{b}}^3$ is odd
Taking a common
$\mathit{a}\left(1+\mathit{a}{\mathit{b}}^3\right)$ is odd
Odd number is obtained after multiplication if both numbers multiplied are odd.
Hence, $\mathit{a}$ is odd and $1+\mathit{a}{\mathit{b}}^3$ is also odd.
$1+\mathit{a}{\mathit{b}}^3$ is odd, so $\mathit{a}{\mathit{b}}^3$ will be even. (odd-1= even quantity)
Because a is odd so for $\mathit{a}{\mathit{b}}^3$ to be even $\mathit{b}$ must be even.
So, $\mathit{a}$ is odd and $\mathit{b}$ is even.