smc

We will use complementary counting to find the probability that the product is not divisible by $4$. Then, we can find the probability that we want by subtracting this from 1. We split this into $2$ cases. Case 1: The product is not divisible by $2$. We need every number to be odd, and since the chance we roll an odd number is $\frac{1}{2},$ our probability is ${\left(\frac{1}{2}\right)}^6=\frac{1}{64}.$ Case 2: The product is divisible by $2$, but not by $4$. We need $5$ numbers to be odd, and one to be divisible by $2$, but not by $4$. There is a $\frac{1}{2}$ chance that an odd number is rolled, a $\frac{1}{3}$ chance that we roll a number satisfying the second condition (only $2$ and $6$ work), and $6$ ways to choose the order in which the even number appears. Our probability is ${\left(\frac{1}{2}\right)}^5\left(\frac{1}{3}\right)\cdot 6=\frac{1}{16}.$ Therefore, the probability the product is not divisible by $4$ is $\frac{1}{64}+\frac{1}{16}=\frac{5}{64}$. Our answer is $1-\frac{5}{64}=\boxed{\frac{59}{64}}$.