Question

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 $4?$

$\dfrac{3}{4}$

$\dfrac{57}{64}$

$\dfrac{59}{64}$

$\dfrac{187}{192}$

$\dfrac{63}{64}$

Solution:

The product fails to be divisible by $4$ when it has at most one factor of $2.$ Each die is odd with probability $\tfrac12,$ contributes exactly one factor of $2$ (a $2$ or $6$ ) with probability $\tfrac13,$ and two factors (a $4$ ) with probability $\tfrac16.$

All six odd: $\left(\tfrac12\right)^6 = \tfrac{1}{64}.$ Exactly one die a $2$ or $6$ and the rest odd: $6 \cdot \tfrac13 \cdot \left(\tfrac12\right)^5 = \tfrac{1}{16} = \tfrac{4}{64}.$

The complement is $\tfrac{1}{64} + \tfrac{4}{64} = \tfrac{5}{64},$ so the answer is $1 - \tfrac{5}{64} = \tfrac{59}{64}.$

Thus, the correct answer is C.

2021 AMC 12B Fall Problem 11

Solution:

Problem 11 in Other Years