<s>[INST] Systems of Congruences \begin{cases} \overline{xyz138} \equiv 0 \mod7 \\ \overline{x1y3z8} \equiv 5 \mod11 \\ \overline{138xyz} \equiv 6 \mod13 \end{cases}
I worked my way up to this:
\begin{cases} 2000x+200y+20z \equiv 2 \mod7 \\ 100000x+1000y+10z \equiv 4 \mod11 \\ 2000x+200y+20z \equiv 7 \mod 13 \end{cases}
I tried to subtract one equation from the other,  but got nothing. No idea what to do.
 [/INST] We have following divisibility rules :

*

*$\overline{abcdef} \equiv \overline{def} - \overline{abc} \pmod 7$

*$\overline{abcdef} \equiv a-b+c-d+e-f \pmod {11}$

*$\overline{abcdef} \equiv \overline{def} - \overline{abc} \pmod {13}$
So one obtains
$$ \overline{xyz} \equiv 5 \pmod 7$$
$$ x+y+z \equiv 6 \pmod {11}$$
$$ \overline{xyz} \equiv 1 \pmod {13}$$
Can you take it from here?
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