Mathematical_Recreations-Kraitchik-2e

Numerical Pastimes 75
Thus, in order that a regular polygon be constructible with
ruler and compass, it is necessary and sufficient that the number
m of its sides be the product by a power of 2 of one of the
above numbers or of a product of prime Fermat numbers at
least one of which is greater than 2 1021 + 1.
7. CYCLIC NUMBERS
We define a cyclic number, or a cyclic set of numbers, as
follows: Let Al = aIlL2' .. a_Ian be an n-digit number in a
positional notation with base B. (We include the possibility
that some of the first digits may be O's.) Then the n numbers
represented by the n cyclic permutations of the digits of AI,
namely AI, A2 = lL2' . ·anal, .•. , An = anal' • ·an-I, are said
to form the cyclic set generated by AI. Thus 142,857 generates
the cyclic set 142,857, 428,571, 285,714, 857,142,
571,428, 714,285 in any scale of notation with base B ~ 9.
If the scale is on the base B =
10 it may be shown that
142,857 is the period in the decimal expansion of +: that is,
t= 0.142857142857142857142···;
and that the succeeding numbers of the cycle are the periods
in the decimal expansion of ~, where x = 3, 2, 6, 4, and 5
respectively.
Bn -1
The sum of the cyclic set AI,' .. , An is S· B-1' where S =
al + ... + an, the sum of the digits, and B is the base of the
scale of notation. It follows from this that if p is a divisor
of ~n ~ ;, the sum of the remainders from division by p of
the n numbers is O.
Also, if one of tnese numbers is divisible
by p, so are the others. For example, in the decimal scale the
numbers 41,205, 12,054, 20,541, 5,412, 54,120 are all divisible
by 41, which is a divisor of 11~ = ~ = 11,111.