<s>[INST] How can I solve this find the digits problem more rigorously? 
Adjoin to the digits $739$ three more digits so that the resulting number $739 \text{_ _ _}$ is divisible by $6, 7, 8$, and $9$.

I can do some quick guess and check as well as some little tricks with $9$, $7$, $6$, and $8$ to arrive to an answer, but how could I show this with more rigor (like more concrete number theory)?
Answers: $739368$ and $739872$
 [/INST] HINT
Let indicate 
$$n=739xyz=7\cdot10^5+3\cdot10^4+9\cdot10^3+x\cdot10^2+y\cdot10+z$$
and consider


*

*$n\equiv 0 \pmod 6 \implies 4+4x+4y+z\equiv 0 \pmod 6\implies 4x+4y+z=2+6k$

*$n\equiv 0 \pmod 7 \implies 3+2x+3y+z \equiv0 \pmod 7 \implies 2x+3y+z=4+7h$

*$n\equiv 0 \pmod 8\implies 4x+2y+z\equiv 0 \pmod 8 \implies 4x+2y+z=8s$

*$n\equiv 0 \pmod 9 \implies 1+x+y+z \equiv 0 \pmod 9\implies x+y+z=8+9t$

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