Detemine the unit digit of a number Find the unit digit of the number:
$$3^{7005} \times 6^{8000}$$
My turn:
$$3^{7005}\times 6^{8000}=3^{7005}\times 3^{8000} \times2^{8000}$$ $$3^{13005} \times 2^{8000}$$
But i could not go on ?

We want the unit digit of the number $3^{7005} \cdot6^{8000}$. This  is equivalent to asking for the residue of the number $3^{7005} \cdot6^{8000}$ modulo $10$. Hence, we want some number $n$ with $0 \leq n \leq 9$ such that
$$
n \equiv 3^{7005} \cdot6^{8000} \pmod{10}
$$
Note that $3^4 \equiv 1 \pmod{10}$ and $6^n \equiv 6 \pmod{10}$ for all $n\in\mathbb{N}$. With this result, we have
\begin{align}
3^{7005} \cdot6^{8000} &= 3^{7004}\cdot 3^{1} \cdot 6^{8000} \\
&= \left(3^{4}\right)^{1751}\cdot 3^{1} \cdot 6^{8000}\\
&\equiv1^{1751}\cdot 3^{1} \cdot 6^{1} \pmod{10} \\
&= 3 \cdot 6 \\
&\equiv 8 \pmod{10} \\
\end{align}
Hence, the unit digit of $3^{7005} \cdot6^{8000}$ is $8$.
