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

Solution 1

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.

$\boxed{\text{E}}$

Solution 2

We are given that $\text{o}\equiv 1\pmod{2}$ , so in mod $2$ we have $1^2+1(n) = n+1$ which is odd only if $\text{n}$ is even $\rightarrow \boxed{\text{E}}$

1986 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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1986 AJHSME Problems/Problem 17

Contents

Problem

Solution

Solution 1

Solution 2

See Also