The only way to get 10 is if one of 13,17, or 19 is paired with 1. WLOG suppose 13 is paired with 1.
13/1
17/19
We have 18 numbers left and want to find the max. number of integers we can obtain. Here is a possible solution.
22/11
21/7
20/10
18/9
16/8
15/5
14/2
12/4
6/3
It is indeed 10.