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

The correct answer is 𝑎 is odd and 𝑏 is even.

To determine which statement is correct, we analyze the parity (whether they are even or odd) of the integers a and b based on the given expression.

We are given that the following expression is odd:

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

We can simplify this by factoring out 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 establish two conditions:

1. a must be odd.
2.
$1+a{b}^3$
must be odd.

Next, we analyze the second condition. Since 1 is an odd number, for the sum
$1+a{b}^3$
to be odd, the term
$a{b}^3$
must be even (because Odd + Even = Odd).

Now, we have determined that the product
$a{b}^3$
is even. Since we already established from the first condition that a is odd, the factor
$b^3$
must be even (since Odd × Even = Even).

For the cube of an integer
$b^3$
to be even, the base integer b itself must be even.

Combining these findings, we conclude that a is odd and b is even.