CONGRUENCE 151

For example, 3467295 = — 4 (mod 7 + 1)
We say “5—9, —4+2, —2—7, —94+6, —3=4,
—74+3, —4"

375, Th.
o= (a+a+a+-)—(atas+as+---) (modr + 1)

In words: If the modulus is the immediate successor of the
radix, any number is congruent to the difference of the sums of its
two sets of alternate digits, the minuend containing the units digit.

This follows from the preceding theorem or from § 366, or
from § 358. It may also be easily proved by definition.

376. As a special case of § 375 we have the following the-
orem :

Th. In the decimal system, if the modulus is eleven, a number
is congruent to the difference of the sums of its two sets of alternate
digits, the minuend containing the units digit.

377. Th. Hence, in the decimal system a number is divisible
by eleven, if, and only if, the difference of the sums of its two sets
of alternate digits, the minuend containing the units digit, 1s so
divisible.

378. We have similar rules for the sexidenal system, with
the modulus seventeen.

379. The modulus 73 4 1 is especially interesting because
in the decimal system 73+ 1 = 1001 =7 X 11 X 13. In
this system therfor the rules stated for the modulus 1001 are
also good for the moduli 7, 11, and 13.

380. Th. If the modulus is r® + 1,
o= o = lag=dw= i |
+ [a1 — (@s — (@7 — -++)) Ir
+ [ao — (a5 — (@5 — +-+))]
For, putting m = 3, po = aor®> + arr + ao § 313.
P = asr® + asr + as
P2 = agr® + arr + as

i

Buta = po — (pr — (p2 — *++)) § 364.