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.