144 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

354. Th. If the modulus is v* — 1, any number, all of whose
digits except two consecutiv digits are zeros, is congruent to the
number obtaind by interchanging these two digits and dropping
or annexing a zero.

For example,

240000 = 42000 = 2400 = 420 = 24 (mod 72 — 1)
(10)00 = (01)0 (mod 72 — 1)

355. Th. If the modulus is r* — 1,

a = ( ak) r + el
k=1 (mod 2) k=0 (mod 2) ;
or a= (2ar(kodd))r + > ai(k even)
or a=(@+as+a+ - )r+(@+at+ar+---)

In words: If the modulus is > — 1, any number is congruent
to the number got by adding to the units digit all the digits that
are alternate to it and. to the digit mext to the units digit those
alternate to it.

Thus, 23,32,43 = 98 (mod 72 — 1)

1,21,22,35 = 79 (mod 72 — 1)

It may assist in this work to join the alternate digits by
loops as below:
21,32.34 = 87 (mod #* — 1)

356. Th. By means of § 355 any number whose digits are all
positiv or zero, or all negativ or zero, and which has more than
two digits may be reduced to a number of not more than two digits
with respect to the modulus r*> — 1.

Thus, if the radix is ¥ and modulus 99,

36,25,47,89 = 1,97 = 98
27,36,87,49 =199 =1,00 =1

If the radix is 2 and modulus 3,
10,11,11,01 = 10,01 = 11