You get dividing the number one and where the remainder is less than 7, places a zero back and cont

When I was a kid, the magic that I liked most was the one where the magician's assistant saw in half. I think grace was trying to understand how he did it, it took me forever to figure out the trick. Using a calculator also have a similar trick, but instead of sawing an assistant, we cut a number two!
142857 * 6 = 857 142 Apparently, the hard part of this trick is to memorize the magic number. When you are surrounded by noisy children, pressure unit conversion chart is not easy to remember 142857! But fortunately, you do not need to memorize the number. Just remember that it is the tithe periodic seventh, and you can use their own calculator to calculate the tithe: 1/7 = 0.142857142857142857 ... The natural question is: has this property with other tithes, or 142857 is special? Amazingly, yes there are other numbers. They have to name: the numbers are cyclical. To find these other numbers, it pays to understand because the seventh works, and it is only to observe the behavior of the tithe in the long division algorithm:
You get dividing the number one and where the remainder is less than 7, places a zero back and continues. Note that when you divide by 7 has only seven remains possible: pressure unit conversion chart 0, 1, 2, 3, 4, 5 and 6. If the remainder is zero at some point, the split ends and the result is accurate. But if at some point the rest repeat, ie, is equal to some else who has appeared before, then you have a tithe. The figures are formed by cyclic maximum pressure unit conversion chart divisions. Because you can never have a zero rest, so in the case of the tithe of 7, the longest period possible pressure unit conversion chart would be six (fortunately is the case). You start with the rest 1, and when it comes in 1 again starts repeating, as in the diagram below:
See how now gives to understand why the numbers work cyclical: 142857 is the tithe of seventh. If we multiply 1/7 by two, we 7/2 and tithe have to be doubled too. But if you look at the diagram, pressure unit conversion chart multiplied by two is the same thing as you start browsing through pressure unit conversion chart the diagram from the second, instead of starting at 1. But no matter where you start, the sequence is always pressure unit conversion chart the same, hence the result will be a rotation of the original tithe!
Knowing that the numbers are cyclical tenths of maximum, you can already start searching for properties of these numbers. What numbers beyond 7 tenths of generate maximum? The first thing we note is that these numbers need to be cousins. The reasoning is fairly simple. Let's call this number k we seek, and do long division of 1 by k. The remains of long division form a recurrence, where the first term is 1, and the following you put a zero at the end and think the rest of the division by k: R [0] = 1 R [n] = 10 * R [ n-1] (mod k) This recurrence gives head to solve: R [n] n = 10 (mod k) To provide a tithe of maximum, the rest needs to be 1 again when n = k-1, ie : R [k-1] = 10 k-1 = 1 (mod k) Now, the Euler-Fermat theorem, we know that: 10 φ (k) = 1 (mod k) where φ (k) is the function Totient. Now we know that when k is composite, the totient is always smaller than k-1, then k can not be made, and therefore is prime. Okay, so k must be prime, but any cousin serves? Nope. Has some cousins that do not work, such as eleven. In the case of 11, it is true that leaves a remainder pressure unit conversion chart of 10 10 1, 10 2 but soon the rest already have one too, so the tithe is much shorter than we would like. In fact, the secret of these cousins that work is ... hum ... nobody knows the secret. This is an open problem. Actually, the thing is so ugly that no one even knows if these primes are finite or infinite. The best we can do is find a script that the first of them: Script in python thinks that the first numbers cyclic After seven, the first cousin who works is 17, and the associated cyclic number is 0588235294117647. Note that this is a case where the leading zero makes no difference! If your calculator has a viewfinder and large, gives to entertain a child for a long time with that number :)
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