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Power and Digit Poser (Posted on 2015-12-23) Difficulty: 3 of 5
Find all possible pairs (M,N) of positive integers such that 10N^2 contains precisely
N*M - M digits.

No Solution Yet Submitted by K Sengupta    
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Solution Analytical solution Comment 2 of 2 |
Starting with Charlie's equation,
m = (n^2 + 1) / (n - 1)

Complete the square to get
m= ((n^2-2n+1) + 2n)/(n-1)
   = (n-1)   + 2n/(n-1)

n-1 is relatively prime to n,
so the only way the right hand side is integral is if
  (n-1) = 1 or (n-1) = 2

So the only solutions are n = 2 or 3.  In both cases, m = 5. 

  Posted by Steve Herman on 2015-12-23 11:34:48
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