24 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Thus @ — s is positiv or negativ according as a is positiv
or negativ.

Nowg=(a —5s):b

Therfor, if @ and b are both positiv or both negativ, g is
positiv; if either @ or b is positiv and the other negativ, g is
negativ.

The converse is proved by the method of exhaustion.

Exs.

 

 

84. Th. Ifb—= 0,a—>> b, f the lower quotient oblaind when
a is divided by b, r the corresponding positiv remainder, and
| | > r, then when a is divided by f, the lower quotient is b and
r the corresponding remainder.

We have a=fb+r

Hence a=0b +7r
The theorem then follows from § 81.
Exs.

 

 

 

85. Th. Ifb—= 0,a—->> b, f the lower quotient oblaind when
a is divided by b, v the corresponding remainder, and |f| > [b],
then when a is divided by f, the lower quotient is b and r the
corresponding remainder.