Question

Sanders 夫人有三个孙辈，他们定期给她打电话。一个每三天打一次，一个每四天打一次，一个每五天打一次。三人都在二千零一十六年十二月三十一日给她打了电话。接下来的一年中，有多少天她没有接到任何孙辈的电话？

Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?

$78$

$80$

$144$

$146$

$152$

答案：D

知识点：容斥原理最小公倍数

难度评级：1800

视频讲解：

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文字解答：

在一个 $60$ 天周期中，第一个孙辈打 $20$ 次，第二个打 $15$ 次，第三个打 $12$ 次，所以 $20 + 15 + 12 = 47$ 。重复统计的天数为 $60 / 12 = 5$ 、 $60 / 15 = 4$ 、 $60 / 20 = 3$ ，因此先得 $47$ ，再减去重复得到 $47 - 5 - 4 - 3 = 35$ 。

第 $60$ 天三人都打电话，被加了三次又减了三次，所以要加回一次，得到每 $60$ 天有 $36$ 天收到电话，也就是 $24$ 天没有电话。二千零一十七年有 $6$ 个完整的 $60$ 天周期，剩下第 $361$ 天和第 $362$ 天没有电话，因此没有电话的天数为 $24 \cdot 6 + 2 = 146$ 。

所以正确答案是 D。

In a $60$ -day period, the first child calls $20$ times, the second child calls $15$ times, and the third child calls $12$ times. $20 + 15 + 12 = 47$ overcounts, however. The first and second children call on the same day $60 / 12 = 5$ times. The first and third children call on the same day $60 / 15 = 4$ times. The second and third children call on the same day $60 / 20 = 3$ times. Subtracting these from $47$ yields $47 - 5 - 4 - 3 = 35.$

The $60$ th day is added in thrice and subtracted out thrice, so we need to add it back in. This means that for every $60$ days, Mrs. Sanders receives a call $36$ days, which means that she does not receive a call on $24$ days. There are $6$ $60$ -day periods, and there are no calls on the $361$ st or $362$ nd day, which results in $24 \cdot 6 + 2 = 146$ total days with no phone calls.

Thus, D is the correct answer.

2017 AMC 8 第 24 题

答案：D

难度评级：1800

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文字解答：

其他年份的第 24 题