122 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

1, 2, ---, the last in the set, » 4 [m| — 1, being evidently
congruent to # — 1, the integer before this to » — 2.

The integers preceding the integer that equals |m| are

rvr+1,r4+2, -4, |m| — 1.
Hence the set
rr+1,r+2,--c,r4+ |m|l —2,7r+ |m| —1

are congruent respectivly to

Lrh L rb2 el —1,0,1,2 0,70 ~27—1,
which are a cyclical arrangement of the integers

0,1,2, «+, |m| — 1.
For example, with respect to the modulus 7,
-3 =2,=10,1,2,3
are congruent to
g .9:60.1.2 3

295. Th. Awny group of |m| successiv integers are congruent,
in order, with respect to the modulus m, to some cyclical arrange-
ment of the set of |m| integers obtaind from the set

0,1,2,3,++, |m| — 3, |m| — 2, |m| —1
by replacing any or all of them by the corresponding integers of
the set
0’ _(lml s ])y _(lml —2)7 _('m} o 3)’ RIS 31 = 2; e

For, if 7 is the positiv remainder obtaind by approximately
dividing a given integer by m, and s the corresponding negativ
remainder, s = — (|m| — 7). § 79.

296. Th. In particular, any group of |m| successiv integers
are congruent, in order, with respect to the modulus m, to the
integers
—a, —(a—1), —(a—2), +--,

_37 _21 _1)07 ]7273y FAFET
lm| —a — 2, |m| —a — 1,

where a is some integer either zero or positiv and less than |m|.