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?

A. 78

B. 80

C. 144

D. 146

E. 152

1. 2017 AMC 8 官方解析

翻译：桑德斯女士有 3 个孙子（女），他们都会定期给她定期打电话．其中一个孙子每 3天打一次电话，另一个孙子每 4天打一次电话，最后一个孙子每 5天打一个电话．3个孙子在 2016年12月 31日这一天都给她打了电话．那么在下一年桑德斯女士有（ ）天没有接到任何一个孙子的电话？
本题容斥原理，$[\frac{365}{3}]=121$，$[\frac{365}{4}]=91$，$[\frac{365}{5}]=73$，$[\frac{365}{12}]=30$，$[\frac{365}{3 \times 5}]=24$，$[\frac{365}{4 \times 5}]=18$，$[\frac{365}{3 \times 4 \times 5}]=6$，所以至少有1名孙子打来电话的天数是：
$121 + 91 + 73 - 30 - 24 - 18 + 6 = 219$（天），无人打来电话的天数是$365 - 219 = 146$（天）．