AMC8 2017

AMC8 2017 · Q24

AMC8 2017 · Q24. It mainly tests Inclusion–exclusion (basic), GCD & LCM.

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. On how many days during the next year did she not receive a phone call from any of her grandchildren?

桑德斯太太有三个孙辈定期给她打电话。一个每 3 天打一次，一个每 4 天打一次，一个每 5 天打一次。他们三人都于 12 月 31 日给她打了电话。接下来一年中，有多少天她没有接到任何孙辈的电话？

(A) 78 78

(B) 80 80

(D) 146 146

(E) 152 152

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): During a 60-day cycle, there are 20 days that the first one calls, 15 days that the second one calls, and 12 days that the third one calls. The sum $20+15+12=47$ overcounts the number of days when more than one grandchild called. There were $60\div 12=5$ days when the first and second called. There were $60\div 15=4$ days when the first and third called. There were $60\div 20=3$ days when the second and third called. Subtracting $5+4+3$ from $47$ leaves $35$ days. But the 60th day was added in three times and subtracted out three times, so there were 36 days in which she received at least one phone call. Thus, in each 60-day cycle, there were $60-36=24$ days without a phone call. In a year, there are six full cycles. Additionally, she receives no phone call on the 361st or 362nd day. Therefore, the total number of days that she does not receive a phone call is $6\cdot 24+2=146$ days.

答案 (D)：在一个 60 天的周期内，第一位孙辈打电话的天数为 20 天，第二位为 15 天，第三位为 12 天。和为 $20+15+12=47$，这会对多位孙辈同一天打电话的天数进行重复计数。第一和第二同日打电话为 $60\div 12=5$ 天；第一和第三为 $60\div 15=4$ 天；第二和第三为 $60\div 20=3$ 天。从 $47$ 中减去 $5+4+3$，剩下 $35$ 天。但第 60 天被加了三次又减了三次，因此她在 36 天里至少收到过一次电话。于是，在每个 60 天周期内，没有电话的天数为 $60-36=24$ 天。一年有六个完整周期。另外，在第 361 天和第 362 天她也没有电话。因此，她一年中没有接到电话的天数总计为 $6\cdot 24+2=146$ 天。

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