If
i.
ASK=(A+S+K)^{3} &
ii. SOLVE=(S+O+L+V+E)^{3} Show that:
i. has one integer solution
and
ii. has several, but only one with 5 distinct digits.
List all of them.
i.
The digital sum of a threedigit number is between 1 and 27.
Only five of these numbers when cubed are threedigit 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 cuberoot of 512 equals the digital sum of 512.
ii.
The digital sum of a fivedigit number is between 1 and 45.
Only twentyfour of these numbers when cubed are fivedigit 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 cuberoots 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 20200817 07:28:39 