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.

Back to the puzzle