Solution
= Solution
Modulo $33$,
$$
10^2=100\equiv1,
\qquad
10\not\equiv1,
$$
so $\operatorname{ord}_{33}(10)=2$. This order is even and
$$
10^{2/2}=10\not\equiv-1\pmod{33}.
$$
The <factor extraction from an even modular order> therefore gives
$$
\gcd(10-1,33)=3,
\qquad
\gcd(10+1,33)=11.
$$
Thus
$$
\boxed{33=3\cdot11.}
$$