jmc

Let $N$ be the product of all odd integers between 0 and 12. Thus, $N=1\times 3\times 5\times 7\times 9\times 11=5(1\times 3\times 7\times 9\times 11)$. The product of odd integers is odd, and the units digit of 5 times any odd number is $5$. Therefore, the units digit of $N$ is $\boxed{5}$.