We write \{a, b, c\} for the set of three different positive integers a, b, and c. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of \{a, b, c\}. We can then calculate the sum of the elements of each subset. For example, for the set \{4,7,42\} we will find sums of 4,7,42,11,46,49, and 53 for its seven subsets. Since 7,11 , and 53 are prime, the set \{4,7,42\} has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, 1 and themselves. In particular, the number 1 is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set \{a, b, c\} that has that number of subsets with prime sums, and explain why no other three-element set could have more.