GATE2018 ME-1: GA-7

Question

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

Answer 1

Given that $a$ and $b$ are integers.

And $a+a^{2}b^{3}$ is odd

So, $a(1+ab^{3})$ is also odd.

We know that only odd$\times$ odd is odd. (even $\times$ even = even, odd $\times$ even = even)
Therefore $a$ is odd and  $(1+ab^{3})$ is odd

So, $ab^{3}=$ even        $\{\because 1+x \text{ is odd}\implies x \text{ is } odd-1\implies x \text{ is even}\}$

Since $a$ is odd and $ab^{3}$ is even, $b$ must be even.

Therefore, $a$ is odd and $b$ is even.

Hence, the correct answer is (D).

Answer 2

Such type of questions can be easily done using example like a = 1 and b = 2 answere will come out to be odd hence a should be odd and b should be even