Is the positive integer n the square of an integer? 4n is the square of an integer n^3 is the...

Understanding the Question

We need to determine whether the positive integer \(\mathrm{n}\) is a perfect square - that is, can \(\mathrm{n}\) be written as \(\mathrm{k}^2\) for some integer \(\mathrm{k}\)?

Given Information

• \(\mathrm{n}\) is a positive integer
• This is a yes/no question: either \(\mathrm{n}\) IS a perfect square or \(\mathrm{n}\) IS NOT a perfect square

What We Need to Determine

For this question to have a sufficient answer, we need to be able to definitively say YES (\(\mathrm{n}\) is always a perfect square) or NO (\(\mathrm{n}\) is never a perfect square) based on the given information.

Key Insight

A number is a perfect square when it equals some integer multiplied by itself (like \(4 = 2 \times 2\), \(9 = 3 \times 3\), \(16 = 4 \times 4\), etc.). We'll use pattern recognition by testing specific examples to discover the underlying rules.

Analyzing Statement 1

Statement 1: \(4\mathrm{n}\) is the square of an integer

What This Means

If \(4\mathrm{n}\) is a perfect square, then \(4\mathrm{n} = \mathrm{k}^2\) for some integer \(\mathrm{k}\). Since \(4\) is already a perfect square (\(2^2\)), let's explore what this tells us about \(\mathrm{n}\).

Testing Different Cases

When n IS a perfect square:

• If \(\mathrm{n} = 1\): then \(4\mathrm{n} = 4 = 2^2\) ✓ Works! (And \(\mathrm{n} = 1 = 1^2\) is a perfect square)
• If \(\mathrm{n} = 4\): then \(4\mathrm{n} = 16 = 4^2\) ✓ Works! (And \(\mathrm{n} = 4 = 2^2\) is a perfect square)
• If \(\mathrm{n} = 9\): then \(4\mathrm{n} = 36 = 6^2\) ✓ Works! (And \(\mathrm{n} = 9 = 3^2\) is a perfect square)

When n is NOT a perfect square:

• If \(\mathrm{n} = 2\): then \(4\mathrm{n} = 8\). Is \(8\) a perfect square? No, because \(\sqrt{8} \approx 2.83\) (not an integer)
• If \(\mathrm{n} = 3\): then \(4\mathrm{n} = 12\). Is \(12\) a perfect square? No, because \(\sqrt{12} \approx 3.46\) (not an integer)

The Pattern Revealed

Notice that \(4\mathrm{n}\) is a perfect square only when \(\mathrm{n}\) itself is a perfect square! This makes sense because \(4 = 2^2\), and for the product of two numbers to be a perfect square, both factors need the right "square structure."

Conclusion: Statement 1 guarantees that \(\mathrm{n}\) must be a perfect square.

[STOP - Sufficient!] Statement 1 alone is sufficient - the answer is always YES.

This eliminates choices B, C, and E.

Analyzing Statement 2

Important: We now forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: \(\mathrm{n}^3\) is the square of an integer

What This Means

If \(\mathrm{n}^3\) is a perfect square, then \(\mathrm{n}^3 = \mathrm{r}^2\) for some integer \(\mathrm{r}\). Let's explore what happens when we cube different types of numbers.

Testing Different Cases

When n IS a perfect square:

• If \(\mathrm{n} = 1 = 1^2\): then \(\mathrm{n}^3 = 1^3 = 1 = 1^2\) ✓ Perfect square!
• If \(\mathrm{n} = 4 = 2^2\): then \(\mathrm{n}^3 = 4^3 = 64 = 8^2\) ✓ Perfect square!
• If \(\mathrm{n} = 9 = 3^2\): then \(\mathrm{n}^3 = 9^3 = 729 = 27^2\) ✓ Perfect square!

Here's the pattern: when \(\mathrm{n} = \mathrm{m}^2\), then \(\mathrm{n}^3 = (\mathrm{m}^2)^3 = \mathrm{m}^6 = (\mathrm{m}^3)^2\), which is always a perfect square.

When n is NOT a perfect square:

• If \(\mathrm{n} = 2\): then \(\mathrm{n}^3 = 8\). Is \(8\) a perfect square? No, because \(\sqrt{8} \approx 2.83\)
• If \(\mathrm{n} = 3\): then \(\mathrm{n}^3 = 27\). Is \(27\) a perfect square? No, because \(\sqrt{27} \approx 5.20\)
• If \(\mathrm{n} = 5\): then \(\mathrm{n}^3 = 125\). Is \(125\) a perfect square? No, because \(\sqrt{125} \approx 11.18\)

Why This Pattern Exists

When you cube a number, you're multiplying it by itself three times. For the result to be a perfect square (an even power of all prime factors), the original number must already have the right structure - it must be a perfect square itself.

Conclusion: Statement 2 guarantees that \(\mathrm{n}\) must be a perfect square.

[STOP - Sufficient!] Statement 2 alone is sufficient - the answer is always YES.

The Answer: D

Both statements independently guarantee that \(\mathrm{n}\) must be a perfect square. Each statement alone provides sufficient information to answer the question with a definitive YES.

• Statement 1: If \(4\mathrm{n}\) is a perfect square, then \(\mathrm{n}\) must be a perfect square
• Statement 2: If \(\mathrm{n}^3\) is a perfect square, then \(\mathrm{n}\) must be a perfect square

Answer Choice D: Each statement alone is sufficient.