Final answer:
To find the greatest number of 'nice pairs' that can be formed from 100 distinct positive integers, consider the possible ratios of these numbers and maximize the number of 'nice pairs'. The maximum number of 'nice pairs' is 2607.
Step-by-step explanation:
To find the greatest number of 'nice pairs' that can be formed from 100 distinct positive integers, we need to consider the possible ratios of these numbers. Since the ratio can be either 2 or 3, we can calculate the number of pairs for each ratio and compare them. Let's assume we have x numbers with a ratio of 2, and y numbers with a ratio of 3. The maximum number of 'nice pairs' can be formed when x and y are as large as possible, while still having distinct numbers.
We can write the number of pairs with a ratio of 2 as x*(x-1)/2 and the number of pairs with a ratio of 3 as y*(y-1)/2. To maximize the number of 'nice pairs', we need to maximize the value of (x*(x-1)/2) + (y*(y-1)/2).
Note that x and y must be positive integers and their sum must be less than or equal to 100, since we have 100 distinct positive integers. By trying different values of x and y, we find that the maximum value of (x*(x-1)/2) + (y*(y-1)/2) is obtained when x=44 and y=55. Therefore, the greatest number of 'nice pairs' that can be formed is 44*(44-1)/2 + 55*(55-1)/2 = 1122 + 1485 = 2607.