Find the largest positive integer $n$ such that $n\varphi(n)$ is a perfect square.  ($\varphi(n)$ is the number of integers $k$, $1 \leq k \leq n$ that are relatively prime to $n$)
To find the largest positive integer \( n \) such that \( n\varphi(n) \) is a perfect square, we need to analyze the properties of \( n \) and its Euler's totient function \( \varphi(n) \).

1. **Express \( n \) in terms of its prime factorization:**
   Let \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \), where \( p_i \) are distinct primes and \( e_i \) are their respective exponents.

2. **Calculate \( \varphi(n) \):**
   The Euler's totient function \( \varphi(n) \) for \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is given by:
   \[
   \varphi(n) = n \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \cdots \left(1 - \frac{1}{p_k}\right)
   \]
   Substituting \( n \):
   \[
   \varphi(n) = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \cdots \left(1 - \frac{1}{p_k}\right)
   \]

3. **Form the product \( n \varphi(n) \):**
   \[
   n \varphi(n) = n \cdot \varphi(n) = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \cdot p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \cdots \left(1 - \frac{1}{p_k}\right)
   \]
   Simplifying:
   \[
   n \varphi(n) = p_1^{2e_1} p_2^{2e_2} \cdots p_k^{2e_k} \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \cdots \left(1 - \frac{1}{p_k}\right)
   \]

4. **Check for perfect square condition:**
   For \( n \varphi(n) \) to be a perfect square, each prime factor in the product must have an even exponent. The term \( \left(1 - \frac{1}{p_i}\right) \) introduces a factor that is not an integer, making it difficult for the product to be a perfect square unless \( n \) is very specific.

5. **Consider small values of \( n \):**
   - For \( n = 1 \):
     \[
     \varphi(1) = 1 \quad \text{and} \quad 1 \cdot 1 = 1 \quad \text{(which is a perfect square)}
     \]
   - For \( n = 2 \):
     \[
     \varphi(2) = 1 \quad \text{and} \quad 2 \cdot 1 = 2 \quad \text{(which is not a perfect square)}
     \]
   - For \( n = 3 \):
     \[
     \varphi(3) = 2 \quad \text{and} \quad 3 \cdot 2 = 6 \quad \text{(which is not a perfect square)}
     \]
   - For \( n = 4 \):
     \[
     \varphi(4) = 2 \quad \text{and} \quad 4 \cdot 2 = 8 \quad \text{(which is not a perfect square)}
     \]

6. **Generalize the observation:**
   For any \( n > 1 \), the product \( n \varphi(n) \) introduces factors that prevent it from being a perfect square due to the non-integer terms \( \left(1 - \frac{1}{p_i}\right) \).

Therefore, the largest positive integer \( n \) such that \( n \varphi(n) \) is a perfect square is indeed \( n = 1 \).

The final answer is \( \boxed{1} \).