Question 66
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.
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.