40 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Th. If a is any integer and r an integer numerically greater
than 1, if also qi, where k = 1, is a quotient, exact, or lower or
upper approximate, obtaind by dividing its predecessor in the
series a, qi, Q2, q3, *** by 7, if also ax_y is the remainder corre-
sponding to the quotient g, then all the remainders are numerically
less than r; for some value of n, where n = 0, the value 0 may
be taken for qn.i1; all quotients after gny1 will be zero;

if @ = 0, the first quotient will be zero and all the remainders
will be zero;

if a —= 0, no quotient need be zero and an infinit number of
sets of remainders can be found, the remainder a, in each set
corresponding to the first zero quotient in that set being equal to
g» and therfor not zero;

and, if for convenience we set ro = 1,

n+1

a = Z ak_fl’k—“l
k=1

Go +air + asr* + <o + au ™! 1 aur”

n+l1
=5 G gayr*¢D

k=1
= B ok @l sivbaar? - ar + a0
Example.

 

 

o S . e s A s

 

120 = 0 + (= (= 3) + 2(— 3 + (= (= 3 + 1(— 3"