a) Is it possible to distribute the numbers $1,2,3, \ldots, 10$ into five pairs such that the five sums of the pairs give five different prime numbers?

b) Is it possible to distribute the numbers $1,2,3, \ldots, 20$ into ten pairs such that the ten sums of the pairs give ten different prime numbers?
a) Yes: $1+4=5,3+8=11,5+2=7,7+6=13$, $9+10=19$.

b) The prime numbers that are the sum of two distinct integers among $1, \ldots, 20$ range from 3 to 39, and thus are among $3,5,7,11,13,17,19,23,29,31,37$. The sum of these prime numbers is 195, which is strictly less than $1+2+\cdots+20=210$. Therefore, it is not possible to perform the grouping as stated in the problem.