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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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