158 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

We may shorten the work still further by applying § 200.
Instead of adding up all the digits and then dividing by 9,
which division is of course equivalent to subtracting a certain
number of 9’s from the sum, we may while adding the digits
subtract 9 from the sum every time that the sum reaches or
surpasses 9.

Thus in finding to what one figure number 72846359 is
congruent we may proceed as follows: ‘7 and 2 are 9, subtract
9, result 0; 8 and 4 are 12, subtract 9, result 3; 3 and 6 are 9,
subtract 9, result 0; 3 and 5 are 8; 8 and 9 are 17, subtract 9,
result 8; thus 72846359 = 8 (mod 9).”

We may simplify still further the process by crossing out
any digit that is a 9 and any set of digits whose sum is 9.
Thus, since 74+ 2=9,44+5=09,and 6 + 3 = 9, we have
72846339 = 8 (mod 9).

The process as thus simplified is called casting out the nines.

We have of course a similar method of casting out the
threes.

Thus 2¥36432 = 2 (mod 3)

Depending on the theorem that

a=(@a-+a+a+- ) — (a+a+a+ ---) (mod 11)

we have the method called by Chrystal casting out the
elevens (1).
Thus

SUA268736 = 6 — (2 + 1) = 3 (mod 11)

There are of course similar methods of casting out the
sevens and thirteens, depending on §§ 390, 392, and a rule
can be workt out corresponding to any positiv integer.

When these methods of casting out the nines, threes,
elevens, etc. are used in checking arithmetical work by con-

(*) See Chrystal, 1. c., p. 178,