152 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

381. Th. If the modulus is r® + 1,

aE( P T ak)—i-( e~ ak)r
k=0 (mod 6) k=3 (mod 6) k=1 (mod 6) k=4 (mod 6)

—l—( e e ak>r2
k=2 (mod 6) k=5 (mod 6)

This comes from § 367 by putting m = 3.

382. Th. In the decimal system, if the modulus is 7, 11, or
13, or any other factor of 1001,

a=[a — (as — (ag — ---))]100
+ [a1 — (@a— (a7 —---)) 110
Ry —(og— --+)) ]

This rule is easy to apply mentally.

For example, 367,421,936 = 882 (mod 7)

We say ““ 3 from 4, 1 from 9, 8; 6 from 2, — 4 from 3, 7;
7 from 1, — 6 from 6, © ; answer 87 = 882.”

Some persons however may find it a help to use a scratch
method, in which each figure is canceld as it is subtracted
from another, the remainder being written under the latter,
as follows :

BOTAZIIZ6
14687T

Of course the subtractions need not be performd in the order
mentiond above. For instance, the new subtrahend may
always be the farthest uncanceld figure to the left. Thus in
the above example we may say 3 from 4, 1; 6 from 2, — 4;
7from1, —6;1from9, 8; — 4 from 3, 7; — 6 from 6, T.

.Negativ digits may be avoided in this example as follows :

BOTA2II36
16
p4882

Wesay ““ 3 from4,1;6 from 12, 6; 7 from 61, 54 ; 5 from 93,
88; 4 from 6, 2" (V).

(1) See William F. White, A Scrap-Book of Elementary Mathematics,
p. 32, The Open Court Publishing Co., Chicago, 1910.