Let's analyze the problem step by step:

We are given that:

• $n$ is a positive integer.
• $n \times 10^n$ is a square number.

We need to find the lowest possible value of $n$.

Step 1: Express the number

The number given in the problem is $n \times 10^n$. We need this to be a square number.

Step 2: Factorize $10^n$

We can express $10^n$ as: $10^n = (2 \times 5)^n = 2^n \times 5^n$ Thus, we have: $n \times 10^n = n \times 2^n \times 5^n$

Step 3: Conditions for a square number

For $n \times 2^n \times 5^n$ to be a perfect square, all the prime factors (including $n$) must occur an even number of times. Therefore, we need to find the smallest value of $n$ such that the prime factorization of $n$ combined with $2^n \times 5^n$ results in every prime factor having an even exponent.

Case 1: Start with $n = 1$

If $n = 1$: $n \times 10^n = 1 \times 10^1 = 10$ But $10$ is not a perfect square.

Case 2: Try $n = 2$

If $n = 2$: $n \times 10^n = 2 \times 10^2 = 2 \times 100 = 200$ But $200$ is not a perfect square.

Case 3: Try $n = 4$

If $n = 4$: $n \times 10^n = 4 \times 10^4 = 4 \times 10000 = 40000$ Now, $40000 = 200^2$, which is a perfect square.

Thus, the smallest possible value of $n$ is $\boxed{4}$.

Would you like further clarification on any part of this solution?

Here are some related questions:

1. What are the properties of a perfect square in prime factorization?
2. Why does $n$ need to have even powers of all prime factors?
3. Can $n \times 10^n$ be a perfect cube instead of a square? What would change?
4. How would you generalize this approach for other powers of 10?
5. Could there be other solutions if we restrict $n$ to be a prime number?

Tip: Always start by checking small values of $n$ when dealing with problems like this, as it can help find patterns faster!