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So close, but no closer... (Posted on 2014-12-04) Difficulty: 3 of 5

Determine the value of the smallest positive integer N such that N1/N contains precisely one zero immediately following the decimal point.

Too easy?

How about seven zeroes, then?

  Submitted by broll    
Rating: 3.0000 (2 votes)
Solution: (Hide)
In general, the mth qualifying number (counting 38 as the zeroeth such number) of the form (n-1)^(1/(n-1)) has the form (as string):

"1.(m zeroes)1(m+2 zeroes)1"; since the last part is so small it can be considered as 1;
1.1001, 1.010001, 1.00100001, 1.0001000001, etc.

This is the form 1+1/(10^(m+1))+1/(10^(2m+4)).

So the function (1+1/(10^(m+1))+1/(10^(2m+4)))^(N-1)+1 = N returns n as the integer part of N.

When m=5, N=16626518.569717461698..., so n = 16626518.
When m=6, N=190660035.08011929935..., so n = 190660035.
When m=7, N= 2148818406.4441059949..., so n= 2148818406.

and so on.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionEasy partMath Man2014-12-08 07:43:32
Some Thoughtsre(3): computer assisted solution, a questionbroll2014-12-05 00:37:57
Questionre(2): computer assisted solution,,,a questionAdy TZIDON2014-12-05 00:18:20
re: computer assisted solutionbroll2014-12-04 23:19:46
Solutioncomputer assisted solutionCharlie2014-12-04 14:21:37
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