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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?

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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!

n is a positive integer. n × 10^n is a square number. What is the lowest possible value of n? You must show your working.

In this problem, we explore the lowest value of n for which n × 10^n is a perfect square. Key concepts include prime factorization and properties of square numbers. Suitable for students in Grades 9-12.

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