Cyclic Numbers
The numbers are cyclic those multiplied by another number less than or equal to the number of digits that he has, his numbers will repeat cyclically, reaching the end those in front. For example: The first number is the cyclic 142,857. If this number (which has six digits) is multiplied by the numbers 1-6 yields:
2 x 142857 = 285714 (note that 1 and 4 were passed to the final)
X = 3 142 857 428 571 (the first pass to the final)
142 857 571 428 x = 4
5 x 142,857 = 714,285
X = 6 142 857 857 142
If we multiply by 7 we get is 999999. This is no accident. This number (142 857) is the periodic part of the division seventh.
The next issue is the cyclic 0588235294117647. If we multiply this number by the numbers 1 to 16 is the same as the previous. If the multiply by 17 results in 99999999999999999.
These numbers are rare to find. Another curious cracterística these numbers is the way you can get them:
We got a prime number and calculate its inverse (1 / p). If the decimal part is periodic and the period have as many digits as the prime number less than 1, then this is a cyclic number. When we divide 1/7 is obtained 0.142857142857142857. Note that is periodic and the period has six digits.
