Explain why $a \pmod 2$ where $a$ is odd has order 1.
Since $a$ is odd, we have by definition of odd
$$ a = 1 \pmod 2 \tag{i}$$
The order of $a$ modulo 2 is the smallest positive $k$ such that
$$ a^k = 1 \pmod 2 \tag{ii}$$
where $\gcd(2,a)=1$, which is true for all odd $a$ since 2 is prime.
We can see from (i) that $k=1$ is the smallest positive integer that satisfies (ii).
So the order of odd $a$ modulo 2 is 1.