Example 1 Find the units digit of $3^{406}$ when it is written in decimal form.
Solve: According to the problem, we are required to find the smallest non-negative remainder $a$ when $3^{406}$ is divided by 10, i.e., $a$ satisfies
$$3^{406} \equiv a(\bmod 10), \quad 0 \leqslant a \leqslant 9$$

Obviously, we have $3^{2} \equiv 9 \equiv-1(\bmod 10), 3^{4} \equiv 1(\bmod 10)$, and thus
$$3^{404} \equiv 1(\bmod 10)$$

Therefore, $3^{406} \equiv 3^{404} \cdot 3^{2} \equiv 9(\bmod 10)$. So the unit digit is 9.