Q. 7 Given that a and b are integers and
a + a²b³ is odd, which one of the following statements is correct?
Consider the algebraic expression:
a + a²b³
where a and b are integers. We want to determine
when this expression is odd.
Correct Answer: Option (D) — a is odd and b is even
Core Concept: Even–Odd Rules
Integers follow simple parity laws:
- Even + Even = Even
- Odd + Odd = Even
- Even + Odd = Odd
- Even × Anything = Even
- Odd × Odd = Odd
Factor the Expression
Factor the given expression:
a + a²b³ = a(1 + ab³)
For a product to be odd, both factors must be odd:
- a must be odd
- 1 + ab³ must be odd
Correct Case: Option (D)
Assume:
- a is odd
- b is even
Then:
- b³ is even
- a · b³ = odd × even = even
- 1 + ab³ = odd + even = odd
- a(1 + ab³) = odd × odd = odd
Example: a = 1 (odd), b = 2 (even)
1 + 1²·2³ = 1 + 8 = 9 (odd)
Why Other Options Fail
Option (A): a odd, b odd
b³ is odd → ab³ is odd → 1 + ab³ is even → odd × even = even.
Fails.
Option (B): a even, b even
a is even → entire expression is even.
Fails.
Option (C): a even, b odd
a is even → a + anything is even.
Fails immediately.
Parity Truth Table
| a parity | b parity | a² parity | b³ parity | a²b³ parity | 1 + a²b³ parity | a + a²b³ parity |
|---|---|---|---|---|---|---|
| Odd | Odd | Odd | Odd | Odd | Even | Even |
| Odd | Even | Odd | Even | Even | Odd | Odd |
| Even | Odd | Even | Odd | Even | Odd | Even |
| Even | Even | Even | Even | Even | Odd | Even |
Exam Tips
- Check all four parity cases quickly
- Even × anything is always even
- For odd results, every factor must be odd
Final Answer: Option (D) — a is odd and b is even
