CONGRUENCE 135

322. Th. A number is divisible by any factor of the square
of the radix when, and only when, its lowest two-figure period is
so divisible.

323. Th. A number written as ordinarily is divisible by the
square of the radix when, and only when, its lowest two digits
are both zeros.

324. Applying § 315 to the case when the radix is ten, we
get the results stated in the following theorem.
Th. In the decimal system

a = ao (mod 2, 5, or 10)
a = ag + 10a, (mod 2, 5, 10, 4, 20, 25, 50, or 100)
a = ay + 10a, + 100ay (mod 2, 5, 10, 4, 20, 25, 50, 100,

8,40, 125, 200, 250, 500, or 1000)

325. As corollaries to these we get the following results:

Th. In the decimal system a number is even if its units digit
15 even, odd if its units digit is odd ; a number written as ordinarily
is divisible by 5 if, and only if, its units digit is either 0 or 5;
a number 1s divisible by 4 if, and only if, the number made up
of its lowest two digits 1s divisible by 4.

Hence a year whose number does not end in two zeros is
a leap year or not according as the number made up of its
lowest two digits is or is not divisible by 4. Thus 1912 was
a leap year, because 12 >> 4, but 1930 was not, because
30-> 4.

326. Applying § 315 to the case when the radix is sixteen,
which of course is now written 10, we get the results stated
in the following theorem.

Th. In the sexidenal notation

a = ay (mod 2, 4, 8, or 10)
a = ayg + 10a, (mod 2, 4, 8, 10, 20, 40, 80, or 100)
a = ay + 10a, + 100as (mod 2, 4, 8, 10, 20, 40, 80, 100, 200,

400, 800, or 1000)

327. As consequences of this theorem we have for numbers
written in the sexidenal notation tests for divisibility by
various powers of 2.