AMC 8 2017 Problem 24
AMC8 2017 年第 24 题
基本计数原理★★★★☆
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?
Sanders 女士有 3 个定期给她打电话的孙子．其中一个每三天通话一次，一个每四天通话一次，还有一个每五天通话一次。这 3 个孙子在 2016 年 12 月 31 号这天同时给她通了话。问下一年中有多少天她一个电话也没收到？
- A.78
- B.80
- C.144
- D.146
- E.152
正确答案 · Correct Answer
D
解析 · Solution
解法1
常规解法 · 容斥
被 3/4/5 整除的天数，按容斥得
无电话 =365-219=146。
巧解 · 60 天一周期
。一周期内有电话天数 =20+15+12-5-4-3+1=36，无电话 =60-36=24。365=6×60+5，前 6 周期 6×24=144；尾 5 天（第 361–365）逐查仅第 364、365 无电话补 2，合 146。
Solution 1
Elementary · inclusion–exclusion
By inclusion–exclusion, the number of days in 365 divisible by 3, 4, or 5 is
So the number of days with no call is 365-219=146.
Slick · a 60-day cycle
Since , one 60-day cycle has 20+15+12-5-4-3+1=36 call days, leaving 60-36=24 with no call. As 365=6×60+5, the six full cycles give 6×24=144; checking the last 5 days (361–365), only days 364 and 365 have no call, adding 2, for a total of 146.
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