CONGRUENCE 121

Since a = 7, these successors are congruent respectivly to
r+1,r42, -, v+ (Im| —2), 7+ (|m| —1) § 204,

For example, let m = 6.
The set of 6 successiv integers 56, 57, 58, 59, 60, 61 are
congruent respectivly to 2, 3, 4, 5, 6, 7.

203. Def. A cyclical arrangement of any set of things,
given in a certain order, as, for example, the letters a, b, ¢,
d, e, f, is an arrangement obtaind by dividing the set into two
parts at any place and then interchanging the two parts,
keeping the order of the things in each part unchanged.

Thus b, ¢, d, e, f, aand e, f, a, b, ¢, d are cyclical arrangements
of the above set of letters.

The original set unchanged is also said to be a cyclical
arrangement of itself.

204. Th. If m is any given modulus, any set of |m| successiv
integers are congruent in order to some cyclical arrangement of
the integers

0,1,2,3,++, |m| — L

By the preceding theorem our set of integers are congruent
respectivly to

rnr+1,7+2, -, v+ (lm| —2),r+ (|m| — 1),

where 7 is positiv or zero and numerically less than m.
If » = 0, this set of integers is the same as

07 1; 27“'! Iml _29 Iml )

and our theorem is proved.
Ifr>0 |m| <r+ |m|land |m| Z r+ |m| — 1
Since r < |ml|,r +1Z |m]
Hencer +12 |m| Z r+ |m| — 1
That is, |m| is one of the integers after the first in the set

rr+1,r4+2, o, r 4+ |m|l — 2,74 |m| —1

This integer is congruent to 0 and the following integers to