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142857, the six repeating digits of 1/7, , is the best-known cyclic number in base 10.〔("Cyclic number" ), ''The Internet Encyclopedia of Science''〕〔Michael W. Ecker, ("The Alluring Lore of Cyclic Numbers" ), ''The Two-Year College Mathematics Journal'', Vol.14, No.2 (March 1983), pp. 105–109〕〔(Cyclic number ), PlanetMath〕 If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2/7, 3/7, 4/7, 5/7, or 6/7 respectively. == Calculations == : 1 × 142,857 = 142,857 : 2 × 142,857 = 285,714 : 3 × 142,857 = 428,571 : 4 × 142,857 = 571,428 : 5 × 142,857 = 714,285 : 6 × 142,857 = 857,142 : 7 × 142,857 = 999,999 (= 142857 + 857142) If you multiply by an integer greater than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits (ones through hundred thousands) to the remaining digits and repeating this process until you have only the six digits left, it will result in a cyclic permutation of 142857 : 142857 × 8 = 1142856 : 1 + 142856 = 142857 : 142857 × 815 = 116428455 : 116 + 428455 = 428571 : 1428572 = 142857 × 142857 = 20408122449 : 20408 + 122449 = 142857 Multiplying by a multiple of 7 will result in 999999 through this process : 142857 × 74 = 342999657 : 342 + 999657 = 999999 If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number. : 8572 = 734449 : 1422 = 20164 : 734449 − 20164 = 714285 It is the repeating part in the decimal expansion of the rational number 1/7 = 0.. Thus, multiples of 1/7 are simply repeated copies of the corresponding multiples of 142857: : 1 ÷ 7 = 0. : 2 ÷ 7 = 0. : 3 ÷ 7 = 0. : 4 ÷ 7 = 0. : 5 ÷ 7 = 0. : 6 ÷ 7 = 0. : 7 ÷ 7 = = 1 : 8 ÷ 7 = 1. : 9 ÷ 7 = 1. : … In base 10, 142,857 is a Harshad number and a Kaprekar number. 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「142857 (number)」の詳細全文を読む スポンサード リンク
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