Say you want to find the units digit of $\displaystyle \left\lfloor \frac{10^{20000}}{10^{100}+3} \right \rfloor $.

So you are working with mod 10.

But how do we get $\displaystyle -3^{199} \equiv -3^{3}(81)^{49} \equiv -27 \equiv 3( \mod 10) $?

What sequence of steps did we use to get to mod 10?