176 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

451. Th. Hence the sum of two numbers has the same units
digit as the sum of their excesses over 5.

452. Th. ab = cd (mod 10)

That is, the product of two numbers is congruent to the product
of their complements with respect to 10, if the modulus s 10.
§ 439.

453. Th. Hence the product of two numbers has the same
units digit as the product of their complements with respect to 10.

454. Th. ab = ef (mod 5)

That is, the product of two numbers is congruent to the product
of their excesses over 5, if the modulus is 5. § 440.

455. Th.

(10<a+b<20)=0<c+d<10)=0<e+f<10)
§ 442.
Hence in this case ¢ + d and e + f are both non-zero digits.

456. Th. If 10 < a 4 b < 20, the unats digit of @ + b equals
e + f and is the complement of ¢ + d.

For by § 451 the units digit of ¢ 4+ b equals the units digit
of e + f, which equals e + f by § 455, which equals the com-
plement of ¢ + d, since e + f = 10 — (¢ + d).

457. Th. If 10 = a + b < 20,
in particular, if 5=Za<10and 5 = b < 10,
then 0<c+d=10and 0= e+ f < 10. § 445.
Hence in this case e + f is a digit, zero or positiv according
asa + b= 10.

458. Th. The excess rule for addition.

I 10 = a + b < 20,
in particular, if 5=Za<10and 5 = b < 10,
then a + b is a two digit number whose first digit is 1 and whose
second digit is the sum of the excesses of a and b over 5.

This follows from §§ 447, 457.