CONGRUENCE 147

This may be proved, if there is more than one period, by
subtracting the highest period from the following period, this
difference from the next period, if there is another period, and
so on, using § 359.

If we use the sign — for the word “from” (Y), so that

a — b = b = a, the conclusion of this theorem may be stated

8= Purr Pui) o Pacsl+ Pans weees

For example, 26,37,48 =48 — (37 — 26) (mod 72 + 1)
11,23,15,48,69 = 11 — 23) — 15) = 48) = 69 (mod 72 + 1)
235,445,325 = 325 — (445 — 235) (mod 7% 4 1)
In the second of these examples we say

“11 from 23, 12 from 15, 3 from 48, 45 from 69, 24.”

365. Th.
a=po—pr+pe—ps+ - 4+ (— 1)"p, (mod r™ + 1)

or o =3 ((— D) (mod ™ + )

This follows from the preceding theorem or it may be proved
from § 313, using the fact that »» = — 1 (mod ™ + 1)
For example,

11,23,15,48,60 = 69 — 48 + 15 — 23 + 11 (mod 72 + 1)

We say “ 69 — 48, 21 + 15, 36 — 23, 13 + 11, 24.”

The method of § 364 seems preferable to that of this article,
since all the operations of that article are subtractions, whereas
these are additions and subtractions.

Def. If we call pp — pr+ p2o— ps+ -+ + (— 1)"p, an
alternating algebraic sum, we may state this theorem asfollows:

If the modulus is r™ + 1, any number is congruent to the alter-
nating algebraic sum of its m-figure periods, beginning with the
period containing the units digit (?).

(1) See Theory of Integers, p. 37.
(%) See J. W. A. Young, Monographs on Modern Mathematics, p. 327,
New York, Longmans, Green, and Co., 1911.