CONGRUENCE 97

228. The work will generally be shorter still if we always
choose the numerically smallest remainder.
For example, compare the following with the work above.

 

229. If a is positiv or zero and b positiv, we may avoid
negativ numbers entirely by always choosing the remainder
corresponding to the exact or lower approximate quotient,
which is positiv or zero. We have the following theorem.

Th. If a is a positiv or zero integer, b a positiv integer, if we
find a series of numbers 1, re, 73, * such that, starting with ry,
each number in the series

a, by i tolsy e

is the remainder got by dividing the predecessor of the preceding
number by the preceding number, choosing always the exact or
lower approximate quotient, then this series will end with a zero
remainder, all the other numbers in the series being positiv except
a, which is positiv or zero, and a (Q b will be the divisor that gives
the zero remainder.

Example.

2415 =3