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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Transposable integer ： ウィキペディア英語版 Transposable integer The digits of some specific integers permute or shift cyclically when they are multiplied by a number ''n''. Examples are: *142857 × 3 = 428571 (shifts cyclically one place left) *142857 × 5 = 714285 (shifts cyclically one place right) *128205 × 4 = 512820 (shifts cyclically one place right) *076923 × 9 = 692307 (shifts cyclically two places left) These specific integers, known as transposable integers, can be but are not always cyclic numbers. The characterization of such numbers can be done using repeating decimals (and thus the related fractions), or directly. ==General== For any integer coprime to 10, its reciprocal is a repeating decimal without any non-recurring digits. E.g. = 0.006993006993006993... 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. This illustrates that cyclic permutations are somehow related to repeating decimals and the corresponding fractions. The greatest common divisor (gcd) between any cyclic permutation of an ''m''-digit integer and 10''m'' − 1 is constant. Expressed as a formula, :$\backslash gcd\backslash left(N,10^m-1\backslash right)=\backslash gcd\backslash left(N\_c,10^m-1\backslash right),$ where ''N'' is an ''m''-digit integer; and ''Nc'' is any cyclic permutation of ''N''. For example, gcd(091575, 999999) = gcd(32×52×11×37, 33×7×11×13×37) = 3663 = gcd(915750, 999999) = gcd(157509, 999999) = gcd(575091, 999999) = gcd(750915, 999999) = gcd(509157, 999999) If ''N'' is an ''m''-digit integer, the number ''Nc'', obtained by shifting ''N'' to the left cyclically, can be obtained from: :$N\_c\; =\; 10\; N\; -\; d\backslash left(10^m-1\backslash right),\; \backslash ,$ where ''d'' is the first digit of ''N'' and ''m'' is the number of digits. This explains the above common gcd and the phenomenon is true in any base if 10 is replaced by ''b'', the base. The cyclic permutations are thus related to repeating decimals, the corresponding fractions, and divisors of 10''m''−1. For examples the related fractions to the above cyclic permutations are thus: *, , , , , and . Reduced to their lowest terms using the common gcd, they are: *, , , , , and . That is, these fractions when expressed in lowest terms, have the same denominator. This is true for cyclic permutations of any integer. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Transposable integer」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.