116 Interesting Properties of Numbers. 



The following fractions, ^V, sVj 2V, 4Vj tVj aV; Aj ^"^^ qVj 

 when expressed in decimals, ^yill give similar results. 



If P = 101 the quotients will be < ^ q, the remainders s qi' 1 



If P = 103 we find ?^p_-j =P— 1 .'. the number of terms in 



6 



P — 1 



the periods is _- — =34 ; which will satisfy equations (9) and ( 12). 



P — 1 



If P — 107 Ave find rr> - =1 .•. the number of terms is — -— 



2 

 not subject to the conditions of (9) and (12. 



If P=109 we find ^p_i=P— 1 •'• the number of terms is 



~2~ 

 P — 1 subject to the conditions of (9) and (12). 



If P=137 we find ?'p_^=P— 1 .*. the number of terms is 



34 



P — 1 



=8 which are subject to the conditions of (9) and (12). 



If P=139 we find ?"p_i=P- 1- •'• the number of terms is 



P — 1 



subject to the conditions of (9) and (12). 



P — 1 



If P=719 we find rr> 1 =1 .'• the number of terms is 



P- 1 2 



2 ■ 



not subject to the conditions of (9) and (12). 



If P have the following values, 113, 131, 503, and 863, we 



shall find ^p _ -1 =P— 1, so that in each case the number of terms 



is P— 1, subject to the conditions of (9) and (12). 



If P=1019 we proceed in the ordinary way until we obtain 

 the remainders r,=10; r2=100; ^3 = 1000; 7-4=829; r^ = 

 138; rg=361. We then multiply Tg into itself and divide the 

 product by 1019, and find for remainder r, 2=908; multiplying 

 r, 3 into itself and dividing by 1019 we find r, ^=93 ; after the 

 same manner we find r^g =497; rge=411; r,g3=786; r^^^ 

 =282; r4 3,=755; r5„, = 923; r5„=S05; r,„3=rp_j = 



