JOMO 5, Short 6

From $7 \nmid xy$ we get $7 \nmid x$ and $7 \nmid y$

From $kx+282y \equiv 0 \mod 7$ we get $kx + 2y \equiv 3kx + 6y \equiv 0 \mod 7$

And from $2x+3y\equiv 0$ we get $4x+6y\equiv 0 \mod 7$

Then: $3kx+6y\equiv 4x+6y \mod 7 \\ 3kx\equiv 4x \mod 7 \\ 3k \equiv 4 \mod 7 \\ k \equiv 6 \mod 7 \\$

$kx+282y=273y+kx+9y \mod(7)=kx+9y \mod(7) \Rightarrow min(k)=\boxed{6}$