Discussion
Given that \(a\) and \(b\) are integers and \(a + a^{2}b^{3}\) is odd, which one of the following statements is correct?
Answer: D
The correct statement is that a is odd and b is even.
If a were even, both a and \(a^{2}b^{3}\) would be even, and their sum would be even. So a must be odd. Then \(a + a^{2}b^{3}\) is odd only if \(a^{2}b^{3}\) is even, and since \(a^{2}\) is odd, \(b^{3}\), and so b, must be even.
Check: a = 1, b = 2 gives 1 + 8 = 9, odd. With both odd, the sum is odd + odd = even; with both even it is even; and with a even the sum is even whatever b is.
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