Determine Parity of Integers $a$ and $b$

We are given that $a$ and $b$ are integers and the expression $a + a^2 b^3$ is odd. Our goal is to determine the parity (odd or even status) of $a$ and $b$.

Analyzing the Expression's Parity

For the sum $a + a^2 b^3$ to be an odd number, the parities of its two terms, $a$ and $a^2 b^3$, must be different. This means one term must be odd and the other must be even.

Step 1: Analyze the Parity of '$a$'

Let's consider the possible parities for $a$:

• If $a$ is even:
  • Then $a^2$ (which is $a \times a$) is also even (even $\times$ even = even).
  • The term $a^2 b^3$ becomes (even $\times b^3$). The product of an even number and any integer ($b^3$) is always even.
  • So, the expression $a + a^2 b^3$ would be (even + even), which results in an even number.
  • This contradicts the given information that $a + a^2 b^3$ is odd. Therefore, $a$ cannot be even.
• Conclusion for $a$: Since $a$ cannot be even, it must be odd.

Step 2: Determine the Parity of '$b$'

Knowing that $a$ must be odd, let's re-examine the expression $a + a^2 b^3$:

• Since $a$ is odd, $a^2$ (which is $a \times a$) is also odd (odd $\times$ odd = odd).
• The expression now looks like: $\text{odd} + (\text{odd} \times b^3)$.
• We know the total expression must be odd. For the sum (odd + term) to be odd, the 'term' must be even. (Because odd + even = odd).
• Therefore, the term $(\text{odd} \times b^3)$ must be even.
• For the product of an odd number and $b^3$ to be even, $b^3$ must be even.
• If $b^3$ is even, then the base integer $b$ must also be even. (If $b$ were odd, $b^3$ would be odd).

Final Conclusion

From our analysis:

• $a$ must be odd.
• $b$ must be even.

This matches the statement '$a$ is odd and $b$ is even'.