Greatest 6 digit number

Relevant wiki: Divisibility Rules

Since the number we are searching for must be divisible by $9$ , we first check $\mod{9}$

$\begin{cases} 9+9+7+9+2+0\equiv 0 \mod{9}\\ 9+9+9+7+6+8\equiv 3 \mod{9}\\ 9+9+9+6+6+0\equiv 3 \mod{9}\\ 9+9+9+7+6+0\equiv 4 \mod{9} \end{cases}$

So the only possible answer is $997920$ , we just need to check modulo $8$ , $5$ and $7$

$997920\equiv 920\equiv 800+120\equiv 120 \equiv 0 \mod{8}$

$997920$ ends with $0$ so it is divisible by 5

$997920\equiv 99792\equiv 9979-4\equiv 9975\equiv 997-10\equiv 987 \equiv 98-14\equiv 84\equiv 0\mod{7}$

Since $LCM(1,2,3,4,5,6,7,8,9,10)=2520$ and $997920+2520=1000440$ , $\boxed{997920}$ is indeed the number we were searchin for