1986 AJHSME Problems/Problem 17

Problem

Let $\text{o}$ be an odd whole number and let $\text{n}$ be any whole number. Which of the following statements about the whole number $(\text{o}^2+\text{no})$ is always true?

$\text{(A)}\ \text{it is always odd} \qquad \text{(B)}\ \text{it is always even}$

$\text{(C)}\ \text{it is even only if n is even} \qquad \text{(D)}\ \text{it is odd only if n is odd}$

$\text{(E)}\ \text{it is odd only if n is even}$

Solution

We can solve this problem using logic.

Let's say that $\text{n}$ is odd. If $\text{n}$ is odd, then obviously $\text{no}$ will be odd as well, since $\text{o}$ is odd, and the product of two odd numbers is odd. Since $\text{o}$ is odd, $\text{o}^2$ will also be odd. And adding two odd numbers makes an even number, so if $\text{n}$ is odd, the entire expression is even.

Let's say that $\text{n}$ is even. If $\text{n}$ is even, then $\text{no}$ will be even as well, because the product of an odd and an even is even. $\text{o}^2$ will still be odd. That means that the entire expression will be odd, since the sum of an odd and an even is odd.

Looking at the multiple choices, we see that our second case fits choice E exactly.

$\boxed{\text{E}}$

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1986 AJHSME Problems/Problem 17

Problem

Solution

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