▁product ▁of ▁three ▁and ▁seven , ▁thus ▁covering ▁the ▁first ▁four ▁prime ▁numbers , ▁many ▁v iges imal ▁fra ctions ▁have ▁simple ▁representations , ▁whether ▁termin ating ▁or ▁rec urr ing ▁( although ▁third s ▁are ▁more ▁complicated ▁than ▁in ▁decimal , ▁repeating ▁two ▁digits ▁instead ▁of ▁one ). ▁In ▁decimal , ▁div iding ▁by ▁three ▁twice ▁( n inth s ) ▁only ▁gives ▁one ▁digit ▁periods ▁( ▁= ▁ 0 . 1 1 1 1 .... ▁for ▁instance ) ▁because ▁ 9 ▁is ▁the ▁number ▁below ▁ten . ▁ 2 1 , ▁however , ▁the ▁number ▁adjacent ▁to ▁ 2 0 ▁that ▁is ▁divis ible ▁by ▁ 3 , ▁is ▁not ▁divis ible ▁by ▁ 9 . ▁N inth s ▁in ▁v iges imal ▁have ▁six - digit ▁periods . ▁As ▁ 2 0 ▁has ▁the ▁same ▁prime ▁factors ▁as ▁ 1 0 ▁( two ▁and ▁five ), ▁a ▁fraction ▁will ▁terminate ▁in ▁decimal ▁if ▁and ▁only ▁if ▁it ▁termin ates ▁in ▁v iges imal . ▁ ▁Cy clic ▁numbers ▁▁ ▁The ▁prime ▁factor ization ▁of ▁twenty ▁is ▁ 2 2   ×   5 , ▁so ▁it ▁is ▁not ▁a ▁perfect ▁power . ▁However , ▁its ▁square free ▁part , ▁ 5 , ▁is ▁con gru ent ▁to ▁ 1 ▁( mod ▁ 4 ). ▁Thus , ▁according ▁to ▁Art in ' s ▁conject ure ▁on ▁primitive ▁roots , ▁v iges imal ▁has ▁infinitely ▁many ▁cy clic ▁pr imes , ▁but ▁the ▁fraction ▁of ▁pr imes ▁that ▁are ▁cy clic ▁is