Question

If \( n^3 \) is odd, which of the following statement(s) is(are) true?
I. \( n \) is odd.
II. \( n^2 \) is odd.
III. \( n^2 \) is even.

• A. I only
• B. II only
• C. I and II
• D. I and III

Hint:

When working with odd and even numbers, remember that the square and cube of an odd number will always be odd, while the square and cube of an even number will always be even.

Answer 1

The Correct Option is C

We are given that \( n^3 \) is odd. Let's analyze the statements one by one: 1. Statement I: \( n \) is odd.
- If \( n^3 \) is odd, then \( n \) must be odd. This is because the cube of an even number is always even. Hence, for \( n^3 \) to be odd, \( n \) must be odd. - Statement I is true. 2. Statement II: \( n^2 \) is odd.
- If \( n \) is odd, then \( n^2 \) will also be odd. This is because the square of an odd number is always odd. Thus, \( n^2 \) is indeed odd if \( n \) is odd. - Statement II is true. 3. Statement III: \( n^2 \) is even.
- This contradicts statement II. If \( n \) is odd, \( n^2 \) cannot be even. Therefore, statement III is false. - Statement III is false. Thus, the correct answer is Option 3: I and II.