5 positive integers must have the following: mean = 5

range = 8

median = 4

mode = 9

Find the numbers and list them in order, starting with the smallest

A mode of 10 means that at least two of the four numbers are 10 (if the other two aren’t also the same, and given that there is no reason to assume that the set has two modes, we will not assume so). So two of the numbers are 10.

The median is eight. Given that there are four numbers in the set, that means that 8 is the average of the two middle numbers. Given that 10 is obviously higher than the median, the third and fourth numbers (when ranked in ascending order) are the two 10′s. The two middle numbers - 10 is one of them - should add up to 16 to produce a median of 8. 6 must be another number in the set.

Finally, knowing that the range is 5, and the largest number is 10, the final number in the set must be 5.

The set of four numbers is thus {5, 6, 10, 10}.