smc

$N$ must be divisible by every positive integer less than $7$, or $1,2,3,4,5,$ and $6$. Each number that is divisible by each of these is a multiple of their least common multiple. $LCM(1,2,3,4,5,6)=60$, so each number divisible by these is a multiple of $60$. The smallest multiple of $60$ is $60$, so the second smallest multiple of $60$ is $2\times 60=120$. Therefore, the sum of the digits of $N$ is $1+2+0=\boxed{3}$