Let $r_1 = 1, r_2 = 3, r_3 = 5, r_4 = 7$, and $a = 3$. Determine the least non-negative residues $x_j$ for
$j= 1, 2, 3, 4$ such that $ar_j ≡ x_j \pmod {8}$.
What do you notice about your results?
The following shows the least non-negative residues for $x_j$ such that $ar_j \equiv x_j \pmod 8$.
| r | $x_j$ |
| 1 | 3 |
| 3 | 1 |
| 5 | 7 |
| 7 | 5 |
Remembering that sets are unordered, we notice that the set of values of $r$ is the same as the set of values of $x_j$.
We further notice that each value is coprime to 8.