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.

Although $8^{4\cdot 2^{n-1}}\equiv 16\pmod {20} $ for all positive integer $n $,  it is actually much simpler:
$8^{4k}\equiv 16\pmod {20} $ for all $k>0$.  That is for any multiple of $4$, not just $4$times powers of $2$.
And furthermore $8^{4k+1}\equiv 16*8\equiv -4*8\equiv-32\equiv 8\pmod {20} $ for $k>0$
And $8^{4k+2}\equiv 8*8\equiv 4\pmod {20} $
And $8^{4k+3}\equiv 4*8\equiv 32\equiv 12\pmod {20} $.
