A cyclic number is an integer of ** N** figures that has the peculiarity that when multiplied by any number between 1 and

**, both inclusive, the result has**

*N***figures, the same as the original cyclic number and in the same cyclic order.**

*N***Find a 6-digit cyclic number.**

#### Solution

The number **142.857** It fulfills this characteristic since:

1 * 142.857 = 142.857

2 * 142.857 = 285.714

3 * 142.857 = 428.571

4 * 142.857 = 571.428

5 * 142.857 = 714.285

6 * 142.857 = 857.142

All results are formed by the same number with the same cyclic order.

All cyclic numbers are the periods of the result of dividing 1 by some prime numbers. Thus, 1/7, generates the unlimited decimal 0.142 857 142 857 142 857. The number of digits for the period is one less than 7. The next prime number that generates a cyclic number is 17 which consists of 16 digits so that if we multiply this number by any other between 1 and 16, both inclusive, the previous 16 digits are obtained in the product and in the same cyclic order.