x = smaller natural number
x+1 = larger natural number 


       x² + (x+1)² = 113
   x² + (x+1)(x+1) = 113
  x² + x² + 2x + 1 = 113
    2x² + 2x - 112 = 0
       x² + x - 56 = 0
    (x + 8)(x - 7) = 0

   x + 8 = 0 ;   x - 7 = 0
       x = -8;       x = 7

Since we are told the answers are natural numbers,
we know that they are positive integers, so we
discard the -8 and 

x = smaller natural number = 7
x+1 = larger natural number = 7+1 = 8

Edwin