Find all numbers of the form p2 × q2 × r2 that are less than 10 000, where
p, q and r are different primes.

To find all numbers of the form \( p^2 \times q^2 \times r^2 \) that are less than 10,000, where \( p \), \( q \), and \( r \) are different primes, let's break down the problem step by step.

### Step 1: Express the number in general form
The numbers we are interested in have the form:
\[
n = p^2 \times q^2 \times r^2 = (pqr)^2
\]
where \( p \), \( q \), and \( r \) are distinct primes.

### Step 2: Determine the maximum value
We need to find the maximum value of \( n = (pqr)^2 \) that is still less than 10,000.

\[
(pqr)^2 < 10,000
\]
\[
pqr < \sqrt{10,000} = 100
\]

### Step 3: Find possible values of \( pqr \)
Now, we consider the smallest primes and calculate their products until the product exceeds 100.

The smallest primes are: 2, 3, 5, 7, 11, 13, ...

Let's calculate the products of combinations of three different primes:

- \( 2 \times 3 \times 5 = 30 \)
- \( 2 \times 3 \times 7 = 42 \)
- \( 2 \times 3 \times 11 = 66 \)
- \( 2 \times 3 \times 13 = 78 \)
- \( 2 \times 5 \times 7 = 70 \)
- \( 2 \times 5 \times 11 = 110 \) (exceeds 100)
- \( 3 \times 5 \times 7 = 105 \) (exceeds 100)
- \( 2 \times 7 \times 11 = 154 \) (exceeds 100)

So, the possible values of \( pqr \) that are less than 100 are:

\[
30, 42, 66, 70, 78
\]

### Step 4: Calculate \( (pqr)^2 \)
For each valid product, calculate \( (pqr)^2 \):

- \( 30^2 = 900 \)
- \( 42^2 = 1764 \)
- \( 66^2 = 4356 \)
- \( 70^2 = 4900 \)
- \( 78^2 = 6084 \)

These are all the numbers of the form \( p^2 \times q^2 \times r^2 \) that are less than 10,000.

### Conclusion
The numbers of the form \( p^2 \times q^2 \times r^2 \) that are less than 10,000 are:
\[
900, 1764, 4356, 4900, 6084
\]

Would you like more details on any of these steps or have any questions?

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**Related Questions:**
1. How do you find the prime factors of a number?
2. What are the properties of prime numbers?
3. How do you calculate the square of a product of numbers?
4. How can the method used above be extended to higher powers?
5. What other forms of numbers can be explored using prime factors?

**Tip:** When working with problems involving prime numbers, always start with the smallest primes and proceed systematically to avoid missing any combinations.

Explore numbers of the form p^2 × q^2 × r^2 less than 10,000, where p, q, and r are distinct primes. Discover the steps and calculations involved in identifying these numbers using prime factorization and squaring. Suitable for high school students studying number theory and prime factorization.

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