552 Mr. W. Shanks on 4872 = 486. [Feb. 15, 



242 08 



is made up of the two factors 243 9s and 100000. .0001. The latter 

 only is divisible by 487. The latter, moreover, admits of the factor 



80 Os ^ 



1000 . . 001 being thrown out, as 487 is not exactly contained in it. We 



81 98 80 Os 



thus have the number 999. .99000. .0001, which may be shown to be 

 divisible by 487^ as follows : — 



Since 15.233, .-.1^^.232; hence ^^^^^0. 

 Therefore ^-486. 



IQIO -|^Q20 IQiO ^Q81 



Again, ^ = 6284, ^ ^ 118602, - 58986, ^ 78153 ; 



io''-i ao''¥io^'— i)-i-i 1 



~^g^ = 78152. Hence = 0, that is = 486. 



We may show that ^^ = 486 in another way, thus : — 



81 98 80 Oa 



Taking, as before, the numbers 99. .99000. .001, and dividing the 

 nines by 487, we have remainder 232. 



Hence, after 232, by dividing by 487, the quotient is 232 times the 



first half+lllf^^xjf>+^=ni). 



Divide first half of quotient by 487, and we have remainder 160. 



160x233 

 Hence — — =268. 



Eemainder from afte^^ 160 is got from "^^^"^^^^"^"^"^"^ ^219 ; hence 

 268 -|- 219 



— -^f^ — = 0. In other words, the quotient obtained from dividing the 



large number (given above), consisting of 162 digits, by 487 is itself 

 divisible by 487. 



It is observable that 486 is an aliquot part of 487^ — 1 ; generally, that 

 P'-lis divisible byP+1. 



The Prime 69499 = 486 ; but, as is usual, 69499' = 486 . 69499. 



81 9s 80 Os 



Hence the number 999 . . 9000 . . 01 is divisible by 69499. We append 

 the results of these divisions, as being somewhat curious, the last result 

 being prime or otherwise. 



