154 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

From this result others may be obtaind, using the facts that
3= —3and — 2 =4 (mod 6)

This theorem may also be proved in the same way as the
two preceding theorems, using the facts that

a = Q@ + 10(11 + 10202 + 103a3 + S a8 + 10"(1"

and 10 =4 = — 2 (mod 6)
whence 12=(—22=4=—2
and I0"=4= —2

The author originally proved the theorem in this way and
it was from observing the resulting fact that, with respect to
the modulus 6 (which equals 2 X 3), any number is congruent
to a linear function of the two numbers, a, and s, to which it
is congruent to the moduli 2 and 3 respectivly, that he was
led to the theorem of § 283, of which § 284, used in the first
proof given above, is a corollary.

388. Th. In the decimal system, if the modulus is 12,

aE—3(a0+2a1)—|—4(a0+a1+a2+a3+---)
=ao— 201+ 4+ as+ ---)
For ea=a +a+a+ as;+ -+ (mod 3)
and a = ap + 2a, (mod 4)
Also —1X34+1X4=1

We then use § 284.

From this theorem others may be obtaind, using the facts
that 1= —11,2= - 10,3 = — 9,4 = — 8 (mod 12).

The theorem may also be proved from § 283, using the fact
that 12 == 4 X 6.

389. Th. In the decimal system, if the modulus is 15,

a=06a—5a+a+t+a+ta+- )
=a—S5@a+ata+ )
390. Th. In the decimal system, if the modulus is 7,

aE< 25 ag— Z ak)+3( 2 G— Z ak)
k=0 (mod 6) k=3 (mod 6) k=1 (mod 6) k=4 (mod 6)

+2(

ar— ay )
k=2 (mod 6) k=5 (mod 6)