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Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$? 
Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability
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 two 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 $1-\frac{5}{64}=$ (C) $\frac{59}{64}$. 