4o positio integers with no mode and their mean is equal to 5 . What is the range thers if the nuedian \( =3 \) ?

To find the range of a set of 40 positive integers with the given conditions, we need to analyze the information provided: 1. **Mean = 5**: This means the sum of all integers is \( 40 \times 5 = 200 \). 2. **Median = 3**: Since there are 40 integers, the median will be the average of the 20th and 21st integers when the numbers are arranged in ascending order. For the median to be 3, both the 20th and 21st integers must be 3 or less. 3. **No mode**: This means that no number appears more than once in the set. Given these conditions, we can start constructing a possible set of integers. ### Step 1: Constructing the Set To satisfy the median condition, we can have the first 20 integers as 1, 2, and 3, ensuring that the 20th and 21st integers are both 3. Let's consider the first 20 integers: - We can have 1, 2, and 3 appear multiple times, but since we need no mode, we can only use each number once. Thus, we can choose: - 1, 2, 3 (3 integers) - Then we can choose 4, 5, 6, ..., up to 20 (17 integers). This gives us: - 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 (20 integers). ### Step 2: Finding the Remaining 20 Integers Now we need to find 20 more integers that will keep the mean at 5 and ensure that the total sum is 200. The sum of the first 20 integers (1 to 20) is: \[ \text{Sum} = \frac{20 \times (20 + 1)}{2} = 210 \] Since we need the total sum to be 200, we need to adjust the next 20 integers to bring the total down by 10. ### Step 3: Adjusting the Next 20 Integers To achieve this, we can choose integers that are larger than 20 but ensure that they do not repeat any of the previous integers. For example, we can choose: - 21, 22, 23, ..., 40 (20 integers). The sum of these integers is: \[ \text{Sum} = \frac{20 \times (21 + 40)}{2} = 610 \] ### Step 4: Final Adjustment Now we need to adjust the integers to ensure the total sum is 200. We can replace some of the larger integers with smaller integers while ensuring that the median remains 3 and there is no mode. ### Conclusion: Finding the Range To find the range, we need the maximum and minimum values in our set. If we take the integers from 1 to 20 and then adjust the next integers to be as small as possible while ensuring the mean is 5, we can have: - Minimum = 1 - Maximum = 20 (or higher depending on adjustments) Thus, the range can be calculated as: \[ \text{Range} = \text{Maximum} - \text{Minimum} = 20 - 1 = 19 \] However, if we adjust the integers to ensure the mean is 5, we can have a maximum integer that is higher than 20, leading to a larger range. ### Final Answer The range of the integers can be calculated based on the adjustments made, but it will depend on the specific integers chosen. In conclusion, the range can be determined based on the integers selected, but it will be at least 19 or more depending on the adjustments made to meet the mean requirement. Thus, the final answer for the range is: \[ \text{Range} = 19 \text{ or more, depending on adjustments.} \]