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116 - Interesting Properties of Numbers. 
The following fractions, #3, #3: 25) fs) ay) gs) sry and vn 
when expressed in decimals, will give dill results. 
If P=101 the quotients will be ; : Pr: the remainders ; oa sap 
If P=103 we find Tp_ yee’ . the number of terms in 
il a 
the periods is ae =8d; which will satisfy equations (9) and (12). 
if P=107 we find Tp_ yale . the numberof terms is 8 
2 é 
not subject to the conditions Pa (9) and (12. 
If P=109 we find Cs. j=P- 1 .*. the number of terms is 
sy 
P—1 subject to the conditions of (9) hss (12). 
if. P= 1 we find __,=P—1.-. the number of terms is 
oe" 
oss which are pee to the conditions of (9) and (12). 
if P=139 we find i 1.*. the number of terms is 
6 
subject to the conditions of (9) and (12). 
P-1 
3 
If P=719 we find rp _ y=! .’. the number of terms is ra 
2 
not subject to the conditions of (9) and (12). 
If P have the following values, 113, 131, 503, and 863, we 
shall find r, _ ; =P —1, so that in each case the number of terms 
2 
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 7, =10; r,=100; r,=1000; r,=829; 75= 
13857, =361, We ches: multiply R. inte itself ficid divide the 
product by 1019, and find for remainder r,,=908; multiplying 
r,, into itself and dividing by 1019 we find re _=93: after the 
same manner we findr,,=497; r,,=411; r,,,=786; Tes 
= 2825 7.95799; 7, ,,= 923; r,,,=805; r, oo ="p_1~ 
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