Chapter 4.1, Problem 18E

Suppose that a and b are integers, $a\equiv 11\left(\text{mod19}\right)$ and $b\equiv \text{3}\left(\text{mod 19}\right)$ . Find the integer c with $\text{o}\le c\le \text{18}$ such that

a) $c\equiv 13a(\mathrm{mod}19)$ .

b) $c\equiv 8b\left(\text{mod19}\right)$ .

c) $c\equiv a-b\left(\text{mod}\text{19}\right)$ .

d) $c\equiv 7a+3b\left(\text{mod19}\right)$ .

e) $c\equiv 2a^2+3b^2\left(\text{mod19}\right)$ .

f) $c\equiv a^3+4b^3\left(\text{mod19}\right)$ .