Let $s(m)$ be the digit sum of $m$. It must hold that $$m\equiv s(m) \equiv 17\equiv -1\pmod{9}$$

Because $m$ is a multiple of $17$, let $m=17k$ which means $$17k\equiv -1\pmod{9}$$

Obviously $k=1$ is one solution, therefore its general solution is $k=9n+1$. Try $n=0, 1, 2, 3,\cdots$ finding $m=\boxed{476}$ is the first positve solution that has $s(m)=17$.