jmc

$N=1\times 2\times 3\times 4\times 6\times 7\times 8\times 9$. We find the units digit of $N$ by disregarding the other digits as we progressively multiply: the units digit of $1\times 2$ is 2; the units digit of $2\times 3$ is 6; the units digit of $6\times 4$ is 4; the units digit of $4\times 6$ is 4; the units digit of $4\times 7$ is 8; the units digit of $8\times 8$ is 4; finally, the units digit of $4\times 9$ is 6. Therefore, the units digit of $N$ is $\boxed{6}$.