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The problem states that we have a set of five positive numbers. The known facts are:

- The greatest number is 10.
- The least number is 1.
- The median of the set is 5.
- The arithmetic mean of the set is also 5.

We need to find the greatest possible values for the two remaining numbers.

### Let's break this down step-by-step:

1. **Let the numbers be**: 
   \[
   x_1, x_2, x_3, x_4, x_5
   \]
   arranged in increasing order, where:
   - \(x_1 = 1\),
   - \(x_5 = 10\),
   - \(x_3 = 5\) (since 5 is the median).

   So we have:
   \[
   1, x_2, 5, x_4, 10
   \]
   
2. **The arithmetic mean is 5**: 
   The arithmetic mean of five numbers is:
   \[
   \frac{1 + x_2 + 5 + x_4 + 10}{5} = 5
   \]

   Simplifying the equation:
   \[
   \frac{16 + x_2 + x_4}{5} = 5
   \]
   \[
   16 + x_2 + x_4 = 25
   \]
   \[
   x_2 + x_4 = 9
   \]

3. **Maximizing \(x_4\) and \(x_2\)**: 
   To maximize \(x_4\), we should try to make \(x_2\) as small as possible, but remember both numbers must be positive. The smallest positive integer \(x_2\) can be is 2. 

   If \(x_2 = 2\), then:
   \[
   x_4 = 9 - 2 = 7
   \]

   Therefore, the two remaining numbers are \(x_2 = 2\) and \(x_4 = 7\).

Thus, the greatest possible values for the two unknown numbers are **7 and 2**, which corresponds to option **(C)**.

Would you like further details or have any questions? 

Here are some related questions for further practice:
1. How would the solution change if the arithmetic mean was 6 instead of 5?
2. What if the least number was 2 instead of 1, how would that affect the remaining numbers?
3. Can you find a similar problem involving a set of 4 numbers instead of 5?
4. What is the general formula for the arithmetic mean in any set of numbers?
5. How do we calculate the median for an even-numbered set?

**Tip**: When working with sets involving means and medians, start by organizing the information in ascending order to simplify calculations.

In a set of five positive numbers, the greatest number is 10 and the least number is 1. If 5 is the median of the set and the arithmetic mean of the series is 5, then what are the greatest possible values of the other two numbers?

Learn how to find the greatest possible values in a set of numbers where the arithmetic mean is 5 and the median is also 5. This problem involves concepts of arithmetic mean, median, and simple algebra, suitable for students in Grades 8-10.

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