unsatisfying proof, by exhaustion, but may serve as a hint

| Comment 1 of 2

I'm sure what's wanted is a deductive proof, rather than a proof by testing all the qualifying numbers for their sum of digits. However, a tabulation of how many of each possible sum of digits might give someone a lead on a deductive proof:

sum of digits number of occurrences 12 1 24 64 36 1000 48 4096 52 192 56 228 60 6859 64 228 68 192 72 4096 84 1000 96 64 108 1