Brave Balloonists

Ten mathematicians are flying on a balloon over the Pacific Ocean. When they are crossing the equator, they decide to celebrate by opening a bottle of champagne. Unfortunately, the cork makes a hole in the balloon. Hydrogen is leaking out, and the balloon is now descending. They need to quickly decide who will take the first turn at trying to fix the balloon. To make this decision in a fair way, each of them writes down an integer a_i not less than $1$ and not more than $10000$. Then they calculate a magic number $N$, which is the number of positive divisors of the product a_1 a_2 \ldots a_{10}. For example, the number of positive integer divisors of $6$ is $4$ (they are $1$, $2$, $3$, and $6$). The first person to act is determined by the last digit of $N$. Your task is to find this digit.