30. Consider the following ten positive integers: If the mean of the above positive integers is 6 , which of the following must be true? I. Median of the ten integers \( =6 \) A. III. Range of the ten integers \( =11 \) B. II only C. I and III only D. II and III only

The mean of the ten positive integers being 6 tells us that the total sum of these integers is \(10 \times 6 = 60\). However, the mean alone doesn't necessarily dictate the median to be 6; the integers could be skewed in such a way that the median is different. For the range, if the integers vary widely and still sum to 60, then the range can be much larger or smaller than 11, depending on how they are distributed. Thus, neither statement is require to be true based solely on the mean provided. In conclusion, neither I nor III must be true, which would lead us to select option B (II only). However, there is not enough information provided to establish the truth of any statement definitively with just the mean. If you pick your integers wisely, you can completely control both the median and the range! For example, having six 5's and four 2's gives you a median of 5 and a range of 4, all while maintaining a mean of 6. The key here is that there are countless combinations that could fit the criteria, making number-crunching both a challenge and a joy!