A new December and a new bunch of puzzles from mscroggs.co.uk.

The only way to reach the sum 16 with 5 different positive integers is 1+2+3+4+6.

The lowest possible sum with 5 different positive integers is 1+2+3+4+5 = 15. In order to reach 16, one of these integers has to be 1 higher. If the integers are still different after this operation, the only candidate is 5.

Alternate proof: Play Killer Sudoku.

The product is 1*2*3*4*6 = 144.