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Verified Answer :

23 and 39

23 + 39 = 62 
39 - 23 = 16

39 and 23!

If we broke down the problem to 62=x+y (y being the smaller number) we can solve from there. 
If y is the smaller number, then y= x-16
Therefore if we plug that back in we get 62=x+x-16 which simplifies to     62=2x-16.
Now we just solve for x which is the bigger number and we get 39. But we still have y which is 16 less so we subtract 16 from 39 to get 23.