i.
The digital sum of a three-digit number is between 1 and 27.
Only five of these numbers when cubed are three-digit numbers.
(Four of these five have distinct digits).                
5 ^ 3 = 125 ;  8
6 ^ 3 = 216 ;  9
7 ^ 3 = 343 ; 10
8 ^ 3 = 512 ;  8
9 ^ 3 = 729 ; 18
Only the cube-root of 512 equals the digital sum of 512.

ii.
The digital sum of a five-digit number is between 1 and 45.
Only twenty-four of these numbers when cubed are five-digit numbers:
22 ^ 3 = 10648 ; 19
23 ^ 3 = 12167 ; 17
24 ^ 3 = 13824 ; 18
25 ^ 3 = 15625 ; 19
26 ^ 3 = 17576 ; 26 =
27 ^ 3 = 19683 ; 27 =
28 ^ 3 = 21952 ; 19
29 ^ 3 = 24389 ; 26
30 ^ 3 = 27000 ;  9
31 ^ 3 = 29791 ; 28
32 ^ 3 = 32768 ; 26
33 ^ 3 = 35937 ; 27
34 ^ 3 = 39304 ; 19
35 ^ 3 = 42875 ; 26
36 ^ 3 = 46656 ; 27
37 ^ 3 = 50653 ; 19
38 ^ 3 = 54872 ; 26
39 ^ 3 = 59319 ; 27
40 ^ 3 = 64000 ; 10
41 ^ 3 = 68921 ; 26
42 ^ 3 = 74088 ; 27
43 ^ 3 = 79507 ; 28
44 ^ 3 = 85184 ; 26
45 ^ 3 = 91125 ; 18
The cube-roots of both 17565 and 19683 are the only two that equal their digital sum.  Of these two, only 19683 is comprised of distinct digits.

Edited on August 17, 2020, 9:04 pm

Posted by Dej Mar on 2020-08-17 07:28:39