Exercise (6.3).9

By using the table you established in the previous question, determine the least non-negative residue $x$ such that

$$ 7^{100}6^{100} \equiv x \pmod {17} $$

The table of indices was calculated as follows.

a mod 17	ind_3(a)
1	16
2	14
3	1
4	12
5	5
6	15
7	11
8	10
9	2
10	3
11	7
12	13
13	4
14	9
15	6
16	8

(a) We use Propositions (6.15) and (6.16) on $7^{100}6^{100} \equiv x \pmod {17}$ to give

$$ \begin{align} 100 \; \text{ind}_3 (7) + 100\text{ind}_3(6) & \equiv \text{ind}_3 (x) \pmod {\phi(17)} \\ \\  100(11 + 15) & \equiv \text{ind}_3 (x) \pmod {16} \\ \\ 2600 & \equiv \text{ind}_3 (x) \pmod {16} \\ \\ \text{ind}_3 (x) & \equiv 8 \pmod{16} \end{align} $$

Using the table of indices, we conclude the least non-negative residue $x$ is 16.