Consider a positive integer X such that
- X and Xn together has 21 digits, and:
- The digital root of (X+Xn) is 1.
Determine the value of n, given that it is a positive integer less than 10.
The first table below lists the digital root of the power shown for a number whose digital root is shown in the first column.
dr power
2 3 4 5 6 7 8 9
1 1 1 1 1 1 1 1 1
2 4 8 7 5 1 2 4 8
3 9 9 9 9 9 9 9 9
4 7 1 4 7 1 4 7 1
5 7 8 4 2 1 5 7 8
6 9 9 9 9 9 9 9 9
7 4 1 7 4 1 7 4 1
8 1 8 1 8 1 8 1 8
9 9 9 9 9 9 9 9 9
But when the original number is added to the power, the new digital root is found in the table below.
dr power
2 3 4 5 6 7 8 9
1 2 2 2 2 2 2 2 2
2 6 1 9 7 3 4 6 1
3 3 3 3 3 3 3 3 3
4 2 5 8 2 5 8 2 5
5 3 4 9 7 6 1 3 4
6 6 6 6 6 6 6 6 6
7 2 8 5 2 8 5 2 8
8 9 7 9 7 9 7 9 7
9 9 9 9 9 9 9 9 9
There are only three 1's in the body of this table:
Either the original number has d.r. 2 and n is 3 or 9, or the original number has d.r. 5 and n is 7.
The cube of 99999 has only 15 digits and adding the length of 5 of 99999 becomes 20, but that of 100000 has 16 digits; together the longer base makes for a total length of 22. So n=3 skips over 21.
Similarly 99^9 has only 18 digits; adding the two from 99 makes it only 20. But 100^9 has 19 digits and the size increase in the number itself makes the total 22. Again a combined length of 21 is skipped over for n=9..
So we're left with needing a d.r. of 5 and an n of 7.
269^7 = 101921535994725989, so the total length is 21.
372^7 = 985826706403442688, likewise.
These are the first and last such satisfactory length totals; they don't have the requisite d.r. 5, but luckily they span more than nine base numbers so all possible digital roots are to be found in the range.
Those that meet the unity criterion are:
X X^7 digital root
(i.e., X^n) of total n+X^n
275 118940252685546875 1
284 149014448675078144 1
293 185384466926009357 1
302 229112403180614528 1
311 281399112371155271 1
320 343597383680000000 1
329 417225924495460409 1
338 503984177369508992 1
347 605767994083541363 1
356 724686190928347136 1
365 863078009304453125 1
So the answer to the puzzle is n = 7.
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Posted by Charlie
on 2022-08-03 08:06:09 |
solution
