Fall 2021 AMC 10 B

The product will not be divisible by 4 precisely when all 6 rolls are odd, or exactly one of them is equal to either 2 or 6 and the rest are odd. The probability of this complementary event is $${\left(\frac{1}{2}\right)}^6+6\cdot \left(\frac{1}{3}\right)\cdot {\left(\frac{1}{2}\right)}^5=\frac{5}{64}.$$ The requested probability is therefore $1-\frac{5}{64}=\frac{59}{64}$.