While the expression of a single series with vinculum on top is adequate, the intention of the above expression is to show that the six cyclic permutations of 006993 can be obtained from this repeating decimal if we select six consecutive digits from the repeating decimal starting from different digits.
The greatest common divisor (gcd) between any cyclic permutation of an m-digit integer and 10m − 1 is constant.
Expressed as a formula, where N is an m-digit integer; and Nc is any cyclic permutation of N. For example, If N is an m-digit integer, the number Nc, obtained by shifting N to the left cyclically, can be obtained from: where d is the first digit of N and m is the number of digits.
There is no solution when n > F. An integer X shift left cyclically by k positions when it is multiplied by a fraction n⁄s.
The direct algebra approach to the above cases integral multiplier lead to the following formula: A long division of 1 by 7 gives: At the last step, 1 reappears as the remainder.
E.g., In this manner, cyclical left or right shift of any number of positions can be performed.
The integer X and its multiple n X then will have the following relationship: which represents the results after left cyclical shift of k positions.
Problem: An integer X shift left cyclically by single position when it is multiplied by 3.
E.g., if an integer X shift right cyclically by single position when it is multiplied by 3⁄2, then 3 shall be the next remainder after 2 in a long division of a fraction 2⁄F.
The following summarizes some of the results found in this manner: A 2-parasitic number 4⁄19, 6⁄19, 8⁄19, 10⁄19, 12⁄19, 14⁄19, 16⁄19, 18⁄19 An integer X shift left cyclically by double positions when it is multiplied by an integer n. X is then the repeating digits of 1⁄F, whereby F is R = 102 - n, or a factor of R; excluding values of F for which 1⁄F has a period length dividing 2 (or, equivalently, less than 3); and F must be coprime to 10.
The following summarizes some of the results obtained in this manner, where the white spaces between the digits divide the digits into 10-digit groups: 1⁄93, 2⁄93, 4⁄93, 5⁄93, 7⁄93, 8⁄93, 10⁄93, 11⁄93, 13⁄93 1⁄87, 2⁄87, 4⁄87, 5⁄87, 6⁄87 In duodecimal system, the transposable integers are: (using inverted two and three for ten and eleven, respectively) Note that the “Shifting left cyclically by single position” problem has no solution for the multiplier less than 12 except 2 and 5, the same problem in decimal system has no solution for the multiplier less than 10 except 3.