Each of the 11 players on the Prime Time soccer team was identified by a different prime number less than 45 on his jersey. The average of those primes was also a prime different from each of those of the players. What were the players' identifying numbers?

There are 14 primes below 45: 2,3,5,7,11,13,17,19,23,29,31,37,41 and 43, and they total 281.

Since we need 11 vests for the soccer team let us first remove the first 3 primes.
This yields a total of 271, being 10 different from 281, and an average of 24.63
If however we remove the last three then the total is 160, being 121 different from 281, with an average of 14.54.

Between 14 and 24 are the primes 17, 19 and 23.
For each of these to be the desired prime average the totals need to be 187, 209 and 253 respectively.  That means respective differences from 281 of 94, 72 and 28.

A difference of 94 is not achievable and it would take 3 odd numbers to reach 91 [7,41 and 43].
A difference of 72 is achievable by removing 2, 29 and 41yielding a total of 209 or an average of 19.
A difference of 28 is achievable by removing 2, 7 and 19 yielding a total of 253 or an avearage of 23.

However the problem states that the average is a different value to the vests.  Removing 2,3 and 23 yields an average of 23 thus the team wears the following vests:
5, 7, 11, 13 17, 19, 29, 31, 37, 41 and 43.

Posted by brianjn on 2009-02-19 18:19:47