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An ''n''-parasitic number (in base 10) is a positive natural number which can be multiplied by ''n'' by moving the rightmost digit of its decimal representation to the front. Here ''n'' is itself a single-digit positive natural number. In other words, the decimal representation undergoes a right circular shift by one place. For example, 4•128205=512820, so 128205 is 4-parasitic. Most authors do not allow leading zeros to be used, and this article follows that convention. So even though 4•025641=102564, the number 025641 is ''not'' 4-parasitic. == Derivation == An ''n''-parasitic number can be derived by starting with a digit ''k'' (which should be equal to ''n'' or greater) in the rightmost (units) place, and working up one digit at a time. For example, for ''n'' = 4 and ''k'' = 7 :4•7=28 :4•87=348 :4•487=1948 :4•9487=37948 :4•79487=317948 :4•179487=717948. So 179487 is a 4-parasitic number with units digit 7. Others are 179487179487, 179487179487179487, etc. Notice that the repeating decimal : Thus : In general, an ''n''-parasitic number can be found as follows. Pick a one digit integer ''k'' such that , and take the period of the repeating decimal ''k''/(10''n''−1). This will be where ''m'' is the length of the period; i.e. the multiplicative order of 10 modulo . For another example, if ''n'' = 2, then 10''n'' − 1 = 19 and the repeating decimal for 1/19 is : So that for 2/19 is double that: : The length ''m'' of this period is 18, the same as the order of 10 modulo 19, so = 105263157894736842. 105263157894736842 × 2 = 210526315789473684, which is the result of moving the last digit of 105263157894736842 to the front. 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「Parasitic number」の詳細全文を読む スポンサード リンク
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