150 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

369. If, instead of subtracting the highest period from the
following period, and so on, as in § 364, we subtract the lowest
period from the preceding period, this difference from the next
period, and so on, we get the following result :

Th. a=[pn— (Pn1— (P2 — -+ — (b2 — (pr — p0)))) I

(mod r™ + 1)

For example,

237,562,431 = [237 — (562 — 431) }*** (mod 7* + 1)

10675 = 106,000,000

This theorem may be seen to agree with that of § 364 by
§ 308.

370. From the theorems of §§ 357-369 we may get special
theorems by puttingm =1, 2, 3, ---
We will mention a few of these.

371. Th. If the modulus is r + 1, any digit may be replaced
by 0 and subtracted from any digit that is an odd number of places
from it or added to any digit that is an even number of places
from it.

372. Th. If the modulus is r + 1, any number, all of whose
digits except ome dre zeros, is congruent to the number obtaind
by changing the sign of that digit and moving it an odd number
of places to the right or left, and is congruent to the number
obtaind by moving it an even number of places to the right or left
without change of sign.

For example, 20000 = 2000 = 200 = 20 = 2 (mod 7 + 1)

373. Th. a=ao— (a1 — (as — (as — -++))) (modr + 1)

For example, 3467295 = — 4 (mod 7 + 1)

We say “ 3 from 4, 1 from 6, 5 from 7, 2 from 2, 0 from 9,
9 from 5, — 4.”

These operations are easily performd mentally.

374. Th. ea=ay—a1+a:—as+ -+ +(— 1)"a, (mod r+ 1)

In words: If the modulus is the immediate successor of the
radix, any number is congruent to the alternating algebraic sum
of its digits, beginning with the units digit.