Determine the order of $32 \pmod {89}$.
The order of $32 \pmod {89}$ exists because $\gcd(89,32)=1$.
Noting that $32=2^5$, we'll first work in base 2 then use the Order Formula.
The order of 2 modulo 89 is a factor of $\phi(89)=88$. These factors are 1, 2, 4, 8, 11, 22, 44, 88 and are the only ones we need to test.
The following calculations show that the order of 2 modulo 89 is 11.
| n | 2^n | 2^n mod 89 |
| 2 | 4 | 4 |
| 4 | 16 | 16 |
| 8 | 256 | 78 |
| 11 | 2048 | 1 |
The Order Formula (6.8) tells us that if $k$ is the order of $a \pmod n$, then the order of $a^s \pmod n$ is $k / \gcd(s,k)$.
And so, the order of $32 \pmod {89}$ is $11 / \gcd(5,11) = 11 / 1 = 11$.