You are requested to partition the set of 10 digits (0,1,…,9) into a maximal number of subsets, such that in each set it is possible to create a prime number
using all its members.

What is the highest prime thus created?

(In reply to Possible solution by Jer)

Yours must be the solution.

To make the maximal number of subsets, we need [2], [3], [5], [7] and then can only create two others, one including the 1 and one including the 9.

To make the largest prime, we need to put as many of the remaining digits in one of these two subsets.  Since neither 1 or 9 is prime, each of these requires at least one more digit.  It can't be the 0, since appending a 0 before or after any non-prime digit won't make it prime.  So the 0 must be in our largest prime, and the optimal place to put it is the tens place.  Of the remaining digits, the largest number (prime or composite) that we can make, subject to the above conditions, is 8609.  Fortunately, this is prime, and we can make a prime from the other two digits (41) so this must be the solution.

Posted by tomarken on 2014-05-16 10:55:39