Sample Question 17:

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?

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

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

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

---

Answer Keys

Question 17: E

---

Solutions

Question 17
Step 1: Assume that \(n\) is an odd number.
- Because \(o\) is also odd, the expression \(no\) will be odd as the product of two odd numbers is odd.
- Because \(o\) is odd, \(o^2\) will be odd too.
- If you add two odd numbers together (which, in this case, are \(o^2\) and \(no\)), you get an even number. So, if \(n\) is odd, the whole expression \(o^2 + no\) is even.

Step 2: Now assume that \(n\) is an even number.
- \(no\) will be even because the product of an even number and an odd number is even.
- \(o^2\) stays odd as before.
- If you add an odd number (\(o^2\)) and an even number (\(no\)), you get an odd number. Therefore, if \(n\) is even, the whole expression \(o^2+no\) is odd.

Step 3: Look at the multiple-choice options and find the one that aligns with the conclusions drawn from step 1 and 2. Option E states that the expression is odd only if \(n\) is even, which matches the conclusion from step 2.

Solution: The correct answer is E.