Determine the orders of the complete residue system modulo 12.
What do you notice about your results?
We first note that 12 is not prime, and so the order only exists for those numbers coprime to 12. These are 1, 5, 7, 11. Let's consider each in turn.
$1^1 \equiv 1 \pmod {12}$, and so the order of 1 modulo 12 is 1.
$5^2 \equiv 1 \pmod {12}$, and so the order of 5 modulo 12 is 2.
$7^2 \equiv 1 \pmod {12}$, and so the order of 7 modulo 12 is 2.
$11^1 \equiv -1 \pmod {12}$, which means $11^2 \equiv 1 \pmod {12}$, and so the order of 11 modulo 12 is 2.