The mean and the median of seven distinct positive integers is 30. If the seven integers are such that their product is maximum, what is the product of the smallest and the largest integers?
*
201
900
180
891

Given that the mean and the median of seven distinct positive integers are both 30, and the goal is to find the product of the smallest and the largest integers such that their product is maximum, we follow these steps:

### Step-by-Step Solution

1. **Mean Calculation:**
   The mean of the seven numbers is given as 30. Therefore, the sum of the seven numbers \(S\) is:
   \[
   S = 7 \times 30 = 210
   \]

2. **Median Determination:**
   Since the median is 30, the fourth number in the sorted list of seven numbers must be 30.

3. **Arranging the Numbers:**
   Let the seven numbers be \(a_1, a_2, a_3, a_4, a_5, a_6, a_7\) such that \(a_1 < a_2 < a_3 < a_4 < a_5 < a_6 < a_7\). Given \(a_4 = 30\), we then have:
   \[
   a_1 + a_2 + a_3 + 30 + a_5 + a_6 + a_7 = 210
   \]
   Simplifying, we get:
   \[
   a_1 + a_2 + a_3 + a_5 + a_6 + a_7 = 180
   \]

4. **Maximizing the Product:**
   To maximize the product of the smallest and the largest integers, we should distribute the remaining sum (180) across the six remaining numbers as evenly as possible, while ensuring they remain distinct positive integers. 

5. **Distributing the Sum:**
   Given the symmetry around the median, we should aim to balance the numbers around 30, using integers close to it. Let's consider:
   \[
   a_1 = 1, a_2 = 2, a_3 = 3
   \]
   Since \(a_4 = 30\), the remaining numbers \(a_5, a_6, a_7\) should satisfy:
   \[
   a_5 + a_6 + a_7 = 180 - (1 + 2 + 3) = 180 - 6 = 174
   \]
   We select \(a_5, a_6, a_7\) such that they are close to each other and above 30.

6. **Possible Choices:**
   By trial, if we choose:
   \[
   a_5 = 40, a_6 = 50, a_7 = 60
   \]
   This gives:
   \[
   1 + 2 + 3 + 30 + 40 + 50 + 60 = 186
   \]
   This sum is not equal to 210. Instead, choose numbers that are even more balanced and add up to 174 and ensure their sum plus 30 is 210:
   \[
   a_1 = 27, a_2 = 28, a_3 = 29, a_4 = 30, a_5 = 31, a_6 = 32, a_7 = 33
   \]
   Now check if this configuration sums to 210:
   \[
   27 + 28 + 29 + 30 + 31 + 32 + 33 = 210
   \]

7. **Product Calculation:**
   With this set, the smallest number \(a_1 = 27\) and the largest number \(a_7 = 33\), the product is:
   \[
   27 \times 33 = 891
   \]

Thus, the product of the smallest and the largest integers, ensuring the numbers are distinct, their sum is 210, and the median is 30, is \( \boxed{891} \).

### Do you want details or have any questions?

### Related Questions:

1. What are the properties of the mean and median in a set of integers?
2. How does balancing around the median help in maximizing or minimizing the product of extremes in a set?
3. Can you explain why the sum of these numbers was set to 210?
4. How would the product change if the integers were not distinct?
5. Can you find another set of numbers that fits the conditions with a different maximum product?
6. What are some other techniques for solving similar maximization problems in statistics?
7. How does ensuring distinct integers affect the possible combinations?
8. Can you explain the significance of balancing numbers around the median in detail?

### Tip:

When faced with optimization problems involving distinct integers, consider the arithmetic mean and distribution properties. Balancing around the median can simplify finding the solution and often leads to the optimal or near-optimal configuration.

Explore how to maximize the product of seven distinct positive integers whose mean and median are both 30. Understand the strategy to distribute numbers around the median to achieve optimal product. Suitable for students in Grades 10-12.

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