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

The correct option is: 𝑎 is odd and 𝑏 is even.

To understand why this is the correct statement, let us analyze the given mathematical expression:

$$a+a^2{b}^3$$

We are given that this expression evaluates to an odd integer. We can simplify the analysis by factoring out the term $a$ from the expression:

$$a(1+a{b}^3)$$

For the product of two integers to be odd, both factors must be odd. Therefore, we can establish two conditions that must be satisfied simultaneously:

1. The first factor, $a$, must be odd.

2. The second factor, $1+a{b}^3$, must also be odd.

Now, let us examine the second condition. If the sum $1+a{b}^3$ is odd, then the term $a{b}^3$ must be even (since an odd number minus 1 is even). Therefore:

$$a{b}^3\text{is even}$$

We already determined from the first condition that $a$ is odd. For the product of two integers $a$ and $b^3$ to be even, at least one of them must be even. Since $a$ is odd, it must be that $b^3$ is even.

Finally, since the cube of an integer preserves its parity (the cube of an odd integer is odd, and the cube of an even integer is even), $b^3$ being even implies that $b$ must be even.

Thus, we have shown that $a$ is odd and $b$ is even.