Given that:

  
  1
------ = 0.000...........0093
 2030           n zeros

Determine the value of n.

Note: Computer program assisted solutions are welcome, but a semi-analytic (simple calculator+p&p) solution is preferred.

*** Adapted from a problem which appeared at the International Mathematics and Science Olympiad in 2020.

log(1/20,10) = -1.301029995663981
10^.301029995663981 = 2
10^(1-.301029995663981) = 5
10^-1.301029995663981 = .05      this has one zero

log(1/20^30,10) =  30 * log(1/20,10) = -39.03089986991943
10^.03089986991943 = 1.0737
10^(1-.03089986991943) = 9.31
10^-39.03089986991943 = decimal point, then 39 zeros, then 93

Posted by Larry on 2023-07-12 07:29:23