▁is ▁a ▁factor ▁of ▁a ▁and ▁a ▁ 3 3 % ▁chance ▁that ▁ 3 ▁is ▁a ▁factor ▁of ▁a , ▁and ▁so ▁on . ▁It ▁can ▁be ▁shown ▁that ▁ 8 8 % ▁of ▁all ▁positive ▁integers ▁have ▁a ▁factor ▁under ▁ 1 0 0 ▁and ▁that ▁ 9 2 % ▁have ▁a ▁factor ▁under ▁ 1 0 0 0 . ▁Thus , ▁when ▁confront ed ▁by ▁an ▁arbitrary ▁large ▁a , ▁it ▁is ▁worth while ▁to ▁check ▁for ▁divis ibility ▁by ▁the ▁small ▁pr imes , ▁since ▁for ▁, ▁in ▁base - 2 ▁. ▁ ▁However , ▁many - digit ▁numbers ▁that ▁do ▁not ▁have ▁factors ▁in ▁the ▁small ▁pr imes ▁can ▁require ▁days ▁or ▁months ▁to ▁factor ▁with ▁the ▁trial ▁division . ▁In ▁such ▁cases ▁other ▁methods ▁are ▁used ▁such ▁as ▁the ▁quadratic ▁s ieve ▁and ▁the ▁general ▁number ▁field ▁s ieve ▁( GN FS ). ▁Because ▁these ▁methods ▁also ▁have ▁super pol ynomial ▁time ▁growth ▁a ▁practical ▁limit ▁of ▁n ▁digits ▁is ▁reached ▁very ▁quickly . ▁For ▁this ▁reason , ▁in ▁public ▁key ▁crypt ography , ▁values ▁for ▁a '' ▁are ▁chosen ▁to ▁have ▁large ▁prime ▁factors ▁of ▁similar ▁size ▁so ▁that ▁they ▁cannot ▁be ▁fact ored ▁by ▁any ▁public ly ▁known ▁method ▁in ▁a ▁useful ▁time ▁period ▁on ▁any ▁available ▁computer ▁system ▁or ▁computer ▁cluster ▁such ▁as ▁super comput ers ▁and ▁computer ▁gr ids . ▁The ▁largest ▁crypt ography - grade ▁number ▁that ▁has ▁been ▁fact ored ▁is ▁R SA - 2 4 0 , ▁a ▁ 2