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Cyclic numbers

Cyclic numbers

A cyclic number is an integer of N figures that has the peculiarity that when multiplied by any number between 1 and N, both inclusive, the result has N figures, the same as the original cyclic number and in the same cyclic order.

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.