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 ?
 A: The unit digit of the numbers of the powers of $3$ form a cyclic sequence: $1,3,9,7,1,3,9,7,\ldots$ In particular, the unit digit of $3^n$ is $3$ if $n$ is of the form $4k+1$. And the unit digit of any power of $6$ is $6$. Since $7\,005$ is of the form $4k+1$, the answer to your question is $8(=3\times6\pmod{10})$.
A: 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 $8$.
A: If you think about the way you typically do multiplication, the unit digit of a product is the product of the unit digits.
Can you figure out the unit digit of the two terms in the product?
Hint: calculate $3^0, 3^1, 3^2, 3^3, 3^4, 3^5,\ldots$  Do you see a pattern in the unit digits?  Do you see why the pattern holds?
