Problems & Puzzles: Puzzles Puzzle 93. Numbers such that the decimal and hexadecimal representation are the reverse of each other The 12/01/2000 Jim Howell wrote:
Solution (1) Jud McCranie wrote "that is all of the solutions. If the number is >=1000000 then it has more decimal digits than hexadecimal digits, so they can't be reversals". But Enoch Haga sent the following additional solution: "8520280 (decimal) = 0820258 (hex)". Maybe Jud was not taking zero leading numbers as valid solutions... (2) Enoch Haga wrote the following argument: "The reason that all of the decimal numbers above are divisible by 53 is that a sum of powers in one base must equal the sum of powers in another base. Example: 5*10^1=50 and 3*10^0=3, sum 53, and in hex: 3*16^1=48 and 5*16^0=5, sum 53. This is the necessary and sufficient condition for reversibility. The first such sum found will not always be prime, but if prime, will divide all subsequent numbers satisfying the same condition because all of the subsequent numbers are multiples of the first prime"... (3) Other solutions of the same type are: 13 (10) = 31 (4) (J & E) 23 (10) =32 (7) (J) 1527465 decimal = 5647251 octal. (J & E) 445 = 5*89 (10) = 544 (9) (J) 315231 (10) = 132513 (12) (J) 43 (13) =34 (13) (J) 834 (10) = 438 (14) (J) 21 (10) = 12 (19) (J & E) 




