5×142857

Five times 142,857 equals 714,285. The six-digit number 142,857 is notable for its cyclic properties and for arising from the decimal expansion of 1/7 in base 10. It also satisfies the relation 7 × 142,857 = 999,999.

142,857 can be obtained as 999,999 divided by 7. The decimal representation of 1/7 is 0.142857142857..., with a repeating block of length six. This period length reflects the multiplicative order of 10 modulo 7, which is 6.

Multiplying 142,857 by 1 through 6 yields six numbers that are cyclic rotations of the original digits: 142,857; 285,714; 428,571; 571,428; 714,285; 857,142. In particular, 5 × 142,857 equals 714,285, which is one of these rotations. This behavior is a hallmark of cyclic numbers in base 10.

Cyclic numbers like 142,857 are closely linked to the concept of full reptend primes for a given base. For base 10, 7 is such a prime, and the corresponding repetend exhibits properties where multiples yield digit rotations. The case of 142,857 and its products provides a classic example of how a single six-digit block can encode symmetric multiplication patterns within the decimal system.