4. Six distinct positive integers $a, b, c, d, e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has 10 prime numbers while Jill claims that she has 9 prime numbers among the sums. Who has the correct claim?
4. Suppose $k$ of the 6 numbers are even. Since the sum of two even or two odd numbers are even, and the sum of two distinct positive positive integers is $>2$, the only even prime, we see that the number of primes among the sums is at most $k(6-k)$. By checking for $k=0,1, \ldots, 6$, we see that the maximum value of $k(6-k)$ is 9 attained when $k=3$. Thus Jack's answer is definitely wrong. Jill's answer is correct because 9 primes can be obtained from the following 6 numbers: $2,4,8,3,15,39$.