How does mod multiplication work? For example, $10^{10} \equiv 4\pmod{6}$
If I used $\pmod{2}$ and $\pmod{3}$, how does the multiplication process work? 
Since $10^{10} \equiv 0 \pmod{2}$ and $10^{10}\equiv 1\pmod{3}$,
$$
10^{10}\equiv (0,1) \pmod{(2,3)}
$$
how do we get the value $4$ at the end? do we list out the possible values of $0\pmod{2}$ and $1\pmod{3}$?
$$
1\pmod{3} = 1, 4, 7, 10
$$
so on.
Since only $4, 10$ and so on satisfy $\pmod{2}$, only values that satisfy both criteria can be used. 
In general, can we do this for $\pmod{n}$, $n$ being any integer?

For integer $n\ge0$
$$10^{n+1}-10=10(10^n-1)\equiv0\pmod{10\cdot9}$$ as $10^n-1$ is divisible by $10-1=9$
$$\implies 10^n\equiv10\pmod{90}\equiv10\pmod6$$
