bad “+> ? 
2 
_ > nteresting Properties of Numbers. 
these quotients and remainders will satisfy the conditions of equa- 
nw (9) and (12). 
e age by the Theory of Numbers, that the remainder of 
a 2 nen by P is either 1 or P—1. Hence, it follows that 
when the remainder is*P —1, the number of terms in the periods 
will be P—1 ora subiitebtipie of P—1. And when the remain- 
Ri or else 
der is 1, the number of terms in the periods must be ya 
a submultiple of 
“at, 
If Nisa oe number of the form x, 6’, 7°, &c. when a, 
8, y, &c. are prime factors, and a, b, c, &c. are whole numbers, 
and P is also a composite number, whose prime factors do not 
differ from those which compose N, then the process will termin- 
ate ; for « can be so taken as to make Ne divisible by P without 
a remainder. : 
If P, besides containing the prime factors common to N, con- 
tains other prime factors, the process will not terminate, but must 
give periods of quotients and remainders; but in this case, other 
terms will occur before the periods commence. 
If Nand Pare both primes, the one of the form 4n+1, and 
the other of the form 4n+3, we know by the law of reciprocily 
of primes, that if the remainder Tp_y is P—1, then also will the 
2 
remainder rj, _ be P—1, when N and P exchange places; 5° 
that the number of terms in the periods in the first case, will be 
P—1; and in the second case, N — 
We will now illustrate these singular properties by numerical 
results. If N=20 and P=37, we shall have as follows: 
; 0,10, 16, 4, 6, 9,14,11,17, 16,15, 2,14, 1, 1,12, 8,17 
Quotients § 49’ 9, 3,15,13,10, 5, 8, 2, 3, 4,17, 5,18, 18, 711, 7 
20, 30, 8, 12, 18, 27, 22, 33, 31, 28, 5, 26, fe "3, 98, 16, 24, 36 
17, 7, 29, 25, 19, 10,15, 4, 6, 9, 32, 11, 35, 34, 14, 21, 13, 1 
We have arranged the quotients in two Soetactita! lines, so that 
the q, is Ainscaly over the ¥p 1 quotient; in this arrang® 
ee 
Remainders i 
ment, we more readily see that they satisfy the condition (1); 
the remainders we have arranged in a similar manner. 
