This question asks for a counterexample to disprove the statement:

"If $n$ is an integer and $n^2$ is divisible by 8, then $n$ is divisible by 8."

Analysis

1. $n^2$ divisible by 8 means $n^2 = 8k$, where $k$ is an integer.
2. To check if $n$ must also be divisible by 8, we will test counterexamples.

Counterexample

Let $n = 4$:

• $n^2 = 4^2 = 16$, and $16$ is divisible by 8.
• However, $n = 4$ is not divisible by 8.

Thus, $n = 4$ disproves the statement.

Would you like more details or additional counterexamples?

Related Questions

1. What is the distinction between divisibility of $n$ and $n^2$?
2. How can we generalize this reasoning for other divisors like 4 or 16?
3. Why does the counterexample $n = 4$ work specifically?
4. Can this logic be applied to higher powers, like $n^3$?
5. How do prime factorization and modular arithmetic help in proving divisibility statements?

Tip

When disproving a statement, always verify the conditions in both directions to ensure the counterexample invalidates the claim.