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Given that $a$ and $b$ are integers and $a+a^2 b^3$ is odd, which of the following statements is correct?

- $a$ and $b$ are both odd
- $a$ and $b$ are both even
- $a$ is even and $b$ is odd
- $a$ is odd and $b$ is even

Migrated from GO Mechanical 3 years ago by Arjun

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Best answer

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

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

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

We now 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).

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

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

We now 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).