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I liked Long Division in school. Is it still taught? There's a neat little trick you can do involving repeating decimals. Ask a friend for a number between 1 and 20. Suppose they pick 5. Have them calculate 5/23. They will start the calculation on the left.
|"STOP!" you say suddenly. If your friend
is really into the division by now, they will probably be quite startled.
You continue, writing the numbers down as you speak. -- "I've done the
whole thing in my head. The next eleven digits are 78260869565, and
then it starts repeating. Check me, if you'd like."
Indeed, 5/23 = .2173913043478260869565 2173913043478260869565 ... This fraction has a repeating decimal, or repetend.
If your friend thinks you've memorized the numbers, you can repeat your trick with 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, or 113. What's the trick? It's all based on a theorem proved in 1836 by E. Midy.
1/97 = 0.
For a prime p, if the repetend of 1/p has 2n digits, then digit (n+k) = 9 - digit (k). So, once your friend has done the hard half of the work, you finish by just subtracting each digit from 9.
For the above numbers, the length of the repetend is p-1. Thus,
you would stop your friend after (p-1)/2 digits. These are known
as long primes, or primes with 10 as a primitive root. Whether there
is an infinite number of these is an unsolved problem.
Ignacio Ruiz de Conejo, Taus Brock-Nannestad, David Fowler, John Bollinger, Dick Saunders Jr, and Moorthy all sent answers. Moorthy also pointed me to http://www.cs.rpi.edu/~moorthy/vm/ , which explains these more.