Chapter 4.1, Problem 17E

Suppose that a and b are integers, $a\equiv 4(\mathrm{mod}13)$ , and $b\equiv 9(\mathrm{mod}13)$ . Find the integer c with $0\le c\le \text{2}$ such that

a) $c\equiv 9a(\mathrm{mod}13)$ .

b) $c\equiv 11b(\mathrm{mod}13)$ .

c) $c\equiv a+b(\mathrm{mod}13)$ .

d) $c\equiv 2a+3b(\mathrm{mod}13)$ .

e) $c\equiv a^{\text{2}}{\text{+b}}^{\text{2}}\left(\text{mod 13}\right)$ .

f) $c\equiv a^{\text{3}}{\text{-b}}^{\text{3}}\left(\text{mod 13}\right)$ .