The statement "If a number $n$ is odd, then $n^2$ is even" is false.

In fact, if $n$ is odd, then $n^2$ will also be odd.

Here’s why:

Proof:

1. An odd number can be expressed in the form $n = 2k + 1$, where $k$ is an integer.
2. Now, square $n$: $n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1$
3. This shows that $n^2$ can be written as $2 \times \text{(an integer)} + 1$, which is the definition of an odd number.

Thus, if $n$ is odd, then $n^2$ will also be odd, not even.

Correct statement:

If $n$ is odd, then $n^2$ is odd.

Would you like further details or clarifications on this concept?

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Here are 5 related questions to explore:

1. What happens if $n$ is even? Is $n^2$ even or odd?
2. Can you prove that the sum of two odd numbers is always even?
3. How does squaring affect other types of numbers, such as rational or irrational numbers?
4. How can we extend the idea of odd and even numbers to modular arithmetic?
5. Is there a pattern in the powers of odd numbers (e.g., $n^3$, $n^4$)?

Tip: When working with odd and even numbers, expressing them in the form $2k$ (even) or $2k + 1$ (odd) helps simplify proofs.