AMC8 2008

AMC8 2008 · Q24

AMC8 2008 · Q24. It mainly tests Probability (basic), Perfect squares & cubes.

Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?

十块编号为1到10的瓷砖面朝下放置。随机翻开一块瓷砖，并掷一颗骰子。瓷砖和骰子上的数字乘积为完全平方的概率是多少？

(A) \frac{1}{10} \frac{1}{10}

(B) \frac{1}{6} \frac{1}{6}

(D) \frac{1}{5} \frac{1}{5}

(E) \frac{7}{30} \frac{7}{30}

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): There are $10\times 6=60$ possible pairs. The squares less than $60$ are $1,4,9,16,25,36$ and $49$. The possible pairs with products equal to the given squares are $(1,1),(2,2),(1,4),(4,1),(3,3),(9,1),(4,4),(8,2),(5,5),(6,6)$ and $(9,4)$. So the probability is $\frac{11}{60}$.

答案（C）：共有 $10\times 6=60$ 种可能的数对。小于 $60$ 的完全平方数为 $1,4,9,16,25,36$ 和 $49$。乘积等于这些平方数的可能数对为 $(1,1),(2,2),(1,4),(4,1),(3,3),(9,1),(4,4),(8,2),(5,5),(6,6)$ 和 $(9,4)$。因此概率为 $\frac{11}{60}$。

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