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Thirteenth power (Posted on 2013-01-12) Difficulty: 4 of 5
The number 21982145917308330487013369 is the thirteenth power of a natural number. Find the number using just pen and paper.

No Solution Yet Submitted by Danish Ahmed Khan    
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re(2): why d1 .?...spoiler | Comment 4 of 7 |
(In reply to re: why d1 .?...spoiler by broll)

I agree that it doesn't seem that easy as not everyone has all the necessary facts on the tip of the tongue, so to speak.

Also regarding logarithms:

In addition to the 13th root being in the high end of the 2-digit numbers and ending in 9, my separation of the candidates 79, 89 and 99 was based on a table of 2-digit common logarithms of 1-digit integers that a keep in my head, partially from memory, partially from combination logic, and partially from the even distribution of the last three:

2 .30
3 .48
4 .60
5 .70
6 .78
7 .85
8 .90
9 .95

Since 79 is close to 80, its log is just under 1.9

  1.9
 x 13
 ----
 24.7

indicating only a 25-digit number and its leading digit is close to 5

89 is close to 90:

 1.95
 x 13
-----
25.35

indicating a 26-digit number with leading digit 2.

and, as Ady mentioned, 99 is way too high, being close to 100^13, with the leading digit certainly exceeding 2.


  Posted by Charlie on 2013-01-12 13:58:49
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