Very interesting. I believe that you should include the original numbers you're multiplying by in the table (in this example they are crossed out because 76 is even).

Without giving away the details to others that want to prove this, the solution involves powers of 2.


Challenge problem:This technique can be generalized  by using powers of n for any other fixed positive integer n>2. How would the algorithm change in this case? (Hint: you need one additional column).