Suppose $77\cdot 88\cdot 99 \equiv n\pmod{20}$, where $0\le n<20$.

What is the value of $n$?
We begin by reducing the factors of the product modulo 20: \begin{align*}
77 &\equiv -3\pmod{20},\\
88 &\equiv 8\pmod{20},\\
99 &\equiv -1\pmod{20}.
\end{align*}(Note that we could have used the more "standard" reductions $77\equiv 17$ and $99\equiv 19$, but the reductions above will make our computations easier.)

Now we have \begin{align*}
77\cdot 88\cdot 99 &\equiv (-3)\cdot 8\cdot(-1) \\
&= 24 \\
&\equiv \boxed{4}\pmod{20}.
\end{align*}