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Fit the middle digit (Posted on 2020-11-12) Difficulty: 3 of 5
Determine all possible digits a for which 9...9a0...09 is a square number. (Excluding the digit a there are 101 9's and 100 0's in the number)

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 5.0000 (1 votes)

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Solution | Comment 1 of 4
Squares that look like this are of the form
(10^n - x)^2 = 10^(2n) - 2x*10^n + x^2

Since the number ends in ...09 we must have x=3.
The first two terms of the square give 10... (2n zeros) minus 60... (n zeros) which will makes the number begin with n 9's followed by a=4. 

(Other endings x^2 give other values of a)

  Posted by Jer on 2020-11-12 08:33:12
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