Cyclic Numbers
Cyclic Numbers
This Demonstration shows the first 10 cyclic numbers.
A cyclic number with digits is such that its digits are shifted cyclically when multiplied by an integer up to . For example, with , ; multiplying by 6 shifts the digits of by three places: . The decimal representation of the reciprocal of has a period of maximum length . So .
n
p
p-1
n=142857
p=7
x
6×n=857142
p
n
p-1
1/7=0.…
142857
For , leading zeros are needed for . For example, the second cyclic number (which comes from ) is the integer .
p>10
n
1/17=0....
0588235294117647
0588235294117647