<s>[INST] 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?
 [/INST] As $\displaystyle10\equiv1\pmod3, 10^n\equiv1\pmod3$ for integer $n\ge0$
As $\displaystyle a\equiv b\pmod m\implies a\cdot c\equiv b\cdot c\pmod{m\cdot c }$ where $a,b,m,c$ are integers
$\displaystyle10\cdot10^n\equiv10\pmod{3\cdot10}\equiv10\pmod6$ as $6|30$
</s>