CONGRUENCE 153

383. Similar scratch methods may be employd for other
moduli. Thus, in the decimal system, adding digits two places
apart,

627346 = 82 (mod 99)
2367
82
In the decimal system,
$27346 = 5 (mod 11)
41 15
X

SPECIAL RULES OF CONGRUENCE
IN THE DECIMAL SYSTEM

384. We will now derive a few special theorems relating to
congruences in the decimal system to small moduli.

385. Th. In the decimal system, if the modulus is 4,
a=a + 2a, = ay — 204
For a = ay + 10a; and 10 = 2 = — 2 (mod 4) § 324.
386. Th. In the decimal system, if the modulus is 8,
a = ay + 2a, + 4a,

For a = ayp + 10a; + 100a; (mod 8) § 324.

And 10 = 2 (mod 8)

Whence 102 = 22

From this result others may be obtaind, using the facts that
1=—-17,2=—6,and 4 = — 4 (mod 8)

387. Th. In the decimal system, if the modulus is 6,
a=3a—2(@+atasta;+--:)=a—2(a1+a+as+---)

For convenience let s = ay + a1 +as +as + -+

Then a = s (mod 3) § 349.
and a = ay (mod 2) § 324.
Also 1 X F L= =1 § 246.

Hence a¢=1 X 3ao+ (— 1) X 25 (mod 2 X 3) § 284.