Mathematics for Computer Science

8.13. References 295

(a)For exactly what integersk > 1is it true that the sum of the digits of the base
16 representation of an integer is congruent modulokto that integer? Justify your
answer.

(b)Give a rule that generalizes this sum-of-digits rule from baseb D 16 to an
arbitrary number baseb > 1, and explain why your rule is correct.

Problem 8.32.
A commutative ring is a setRof elements along with two binary operations ̊and
̋fromRRtoR. There is an element inRcalled the zero-element, 0 , and
another element called the unit-element, 1. The operations in a commutative ring
satisfy the followingring axiomsforr;s;t 2 R:

.r ̋s/ ̋tDr ̋.s ̋t/ (associativity of ̋);

.r ̊s/ ̊tDr ̊.s ̊t/ (associativity of ̊);

r ̊sDs ̊r (commutativity of ̊)

r ̋sDs ̋r (commutativity of ̋);

0 ̊rDr (identity for ̊);

1 ̋rDr (identity for ̋);

9 r^02 R: r ̊r^0 D 0 (inverse for ̊);

r ̋.s ̊t/D.r ̋s/ ̊.r ̋t/ (distributivity):

(a)Show that the zero-element is unique, that is, show that ifz 2 Rhas the
property that
z ̊rDr; (8.35)

thenzD 0.

(b)Show that additive inverses are unique, that is, show that

r ̊r 1 D 0 and (8.36)

r ̊r 2 D 0 (8.37)

impliesr 1 Dr 2.

(c)Show that multiplicative inverses are unique, that is, show that

r ̋r 1 D 1

r ̋r 2 D 1

impliesr 1 Dr 2.