number theory - Modular exponentitation (Finding the remainder)

I want to find the remainder of $8^{119}$ divided by $20$ and as for as i do is follows:

$8^2=64\equiv 4 \pmod {20} \\

8^4\equiv 16 \pmod {20} \\
8^8\equiv 16 \pmod {20}\\
8^{16}\equiv 16 \pmod {20}$

from this i see the pattern as follows $8^{4\cdot 2^{n-1}} \text{is always} \equiv 16 \pmod {20} \,\forall n \ge 1$

So,
$\begin{aligned} 8^{64}.8^{32}.8^{16}.8^7 &\equiv 16.8^7 \pmod{20}\\
&\equiv 16.8^4.8^3 \pmod{20} \\
&\equiv 16.8^3 \pmod {20}\end{aligned}$

And i'm stuck. Actually i've checked in to calculator and i got the answer that the remainder is $12$. But i'm not satisfied cz i have to calculate $16.8^3$
Is there any other way to solve this without calculator. I mean consider my condisition if i'm not allowed to use calculator.

Thanks and i will appreciate the answer.

Answer

You are almost there. You can reduce anything modulo $20$ so:

$$16\times 8^3\equiv 16\times 8^2\times 8\equiv 16\times 4 \times 8\equiv64\times 8\equiv 4\times 8\equiv 12$$

You could also have used $16\equiv -4$ if it had helped - sometimes negative numbers make life easier, but it was not necessary here.