In details

What are cyclic numbers?


Cyclic numbers are those that multiplied by another number less than or equal to the number of digits that it has, their numbers will be repeated cyclically, passing to the end those in front. For example: The first cyclic number is the 142857. If this number (which has six digits) is multiplied by the numbers from 1 to 6 we get:

2 x 142857 = 285714 (note that 1 and 4 were passed to the end)
3 x 142857 = 428571 (1 passes to the end)
4 x 142857 = 571428
5 x 142857 = 714285
6 x 142857 = 857142

If we multiply by 7 what we get is 999999. This is no coincidence. This number (142857) is the periodic part of division 1/7.

The next cyclic number is the 0588235294117647. If we multiply this number by the numbers from 1 to 16, the same happens with the previous one. Multiplying it by 17 results in 99999999999999999.

These numbers are rare to find. Another curious feature of these numbers is how to obtain them:

We take a prime number and calculate its inverse (1 / p). If the decimal part is periodic and the period has as many digits as the prime number minus 1, then this is a cyclic number. When we divide 1/7 we get 0.1442857142857142857. Note that it is periodic and the period has six digits.