Written as a product of its prime factor, 600 = 2^3 x 3 x 5^2. Find the smallest positive integer k such that 600/k is a square number

To solve this problem, we need to find the smallest positive integer \( k \) such that \( \frac{600}{k} \) is a square number.

### Step 1: Prime factorization of 600
We are given that:
\[
600 = 2^3 \times 3 \times 5^2
\]

### Step 2: Representing \(\frac{600}{k}\) as a square number
Let \( \frac{600}{k} \) be a square number. This implies that:
\[
\frac{600}{k} = n^2 \quad \text{for some integer } n
\]
This means \( k \) must remove all odd exponents in the prime factorization of 600, leaving only even exponents so that the result is a perfect square.

### Step 3: Identify the smallest \( k \)
The prime factorization of \( 600 = 2^3 \times 3 \times 5^2 \) contains the following:
- \( 2^3 \) has an odd exponent (3).
- \( 3^1 \) has an odd exponent (1).
- \( 5^2 \) has an even exponent (2).

To make \( \frac{600}{k} \) a perfect square, we need to:
- Remove one factor of 2 (to make \( 2^3 \) an even exponent of \( 2^2 \)).
- Remove the factor of 3 (to make \( 3^1 \) disappear).

So, the smallest \( k \) that satisfies this condition is:
\[
k = 2 \times 3 = 6
\]

### Step 4: Verification
Now, let's check if \( \frac{600}{6} \) is indeed a square number:
\[
\frac{600}{6} = \frac{2^3 \times 3 \times 5^2}{2 \times 3} = 2^2 \times 5^2 = (2 \times 5)^2 = 10^2 = 100
\]
Since \( 100 \) is a perfect square, our solution is correct.

### Final Answer
The smallest positive integer \( k \) such that \( \frac{600}{k} \) is a square number is \( \boxed{6} \).

Would you like more details or have any questions? 

Here are 5 related questions:
1. How would you factorize 900 into its prime factors?
2. What is the smallest \( k \) such that \( \frac{900}{k} \) is a square number?
3. How can you generalize this method to find \( k \) for any number \( N \)?
4. What if we wanted \( \frac{600}{k} \) to be a cube instead of a square?
5. How do you determine if a large number is a perfect square?

**Tip:** Always check the prime factorization carefully to ensure you're removing all odd exponents when looking for perfect squares.

Learn how to find the smallest positive integer k such that 600 divided by k is a square number. This problem involves prime factorization and the concept of perfect squares. Suitable for Grades 7-9 students.

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