AMC 8 2003 Problem 12

AMC8 2003 年第 12 题

概率★★★☆☆

When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the faces that can be seen is divisible by 6 ?

当把一个标准的六面骰子掷到桌面上时，骰子底面数字无法看到。那么其他能看到的五个面上的数字之积能被 6 整除的概率是多少？

A. $\displaystyle \frac{1}{3}$
B. $\displaystyle \frac{1}{2}$
C. $\displaystyle \frac{2}{3}$
D. $\displaystyle \frac{5}{6}$
E.1

正确答案 · Correct Answer

解析 · Solution

Solution 1

We have six cases: each different case, every one where a different number cannot be seen. The rolls that omit numbers one through five are all something times six: an example would be where the number you cannot see is one, so the product should be 2 x 3 x 4 x 5 x 6, and so product should be divisible by six. The roll that omits six on the other hand is 1 x 2 x 3 x 4 x 5, which has 2 x 3, also equal to six. We can see that all of them have a factor of 6 and therefore are divisible by six, so the solution should be

\textbf{(E)}

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