Solution to: Elegant Equation

Make a long division:

4/4.....\......

Calculate the first digit of the result:

4/4.....\1.....
  4
 --
  0

Now we also know that the next digit of the dividend is a 1:

4/41....\1.....
  4
 --
  0

Then we can calculate the next digit of the result:

4/41....\10....
  4
 --
  01
   0
  --
   1

Therefore, the next digit of the dividend is also a 0:

4/410...\10....
  4
 --
  01
   0
  --
   1

This process of calculating the next digit of the result and adding it to the dividend must be repeated until the remainder is 0 and the last digit of the result is a 4:

4/410256\102564
  4
 --
  01
   0
  --
   10
    8
   --
    22
    20
    --
     25
     24
     --
      16
      16
      --
       0

Conclusion: the number we were looking for is 10256.

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