This method is guaranteed (*) to amaze your friends. On the rare occasion you need to divide an integer by seven, with a non-integer result, you can quote the result to arbitrary precision, but usually seven or eight decimal places is the best limit to avoid disclosing your method.
- Firstly, memorize this sequence: 142857.
- Do the integer part of the division, working out the remainder. No tricks here, I'm afraid.
- Now the clever bit. Take the remainder (n) and find the nth smallest digit in the memorized sequence. Starting with that digit, the decimal part of the result is that six-digit sequence repeated over and over.
Let's look at an example that amazed my friends and had them looking for the wires in my brain. We were in a 7-piece band that was potentially going to get paid £200 for a gig (that never happened). I was able to tell them in about two seconds flat that the split amount would be £28.57.
I decided to take out a factor of 100 at the beginning and treat the sum as 2/7, but you could go through the motions to get 28 with a remainder of 4. With 2/7, the second-smallest digit of 142857 is the 2, so I had 0.285714285714..., multiplied by 100 to get £28.57. For 28 remainder 4, the fourth-smallest digit is the 5, to give 28.57142857...
Try not to give too many places, or the recursive nature of the pattern will become clear.
What a fun, exciting life I lead, eh?
(*) Guarantee not applicable. | 
