Question

If 3 divides n - 2, then n is not a perfect square.
1) True
2) False

Answer 1

The statement is false, as it is possible for an integer that meets the condition "3 divides n - 2" to be a perfect square, as shown with the example of n = 11.

Step-by-step explanation:

The statement that “If 3 divides n - 2, then n is not a perfect square” is false. Consider the example where n = 11. Here, 11 - 2 = 9, which is divisible by 3, and 11 is also a perfect square. This counterexample clearly demonstrates that an integer n can be a perfect square even if 3 divides n - 2. To better understand this concept, let's break it down:

• If 3 divides n - 2, this implies that n - 2 = 3k for some integer k.
• We can rearrange this equation to find n = 3k + 2.
• The claim is that n being a perfect square is not possible. However, we found that n can equal a number that is a perfect square (like 11).

Therefore, the correct statement would be: “If 3 divides n - 2, n may or may not be a perfect square.”