2017 AMC 8 Problems/Problem 24
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?
Solution 1 (Least Common Multiple)
Note that so there is a cycle every days.
As shown below, all days in a cycle that Mrs. Sanders receives a phone call from any of her grandchildren are colored in red, yellow, or green. The year 2017 has days, or cycles and days.
- For each cycle, there are days that Mrs. Sanders does not receive a phone call, as indicated by the white squares.
- For the last days, there are days that Mrs. Sanders does not receive a phone call, as indicated by the first days in a cycle.
Together, the answer is
Solution 2 (Detailed Version of Principle of Inclusion-Exclusion)
We use Principle of Inclusion-Exclusion. There are days in the year, and we subtract the days that she gets at least phone call, which is
To this result we add the number of days where she gets at least phone calls in a day because we double subtracted these days. This number is
We now subtract the number of days where she gets three phone calls, which is . Therefore, our answer is
Solution 3 (Condensed Version of Principle of Inclusion-Exclusion)
For this problem, we use the Principle of Inclusion-Exclusion (PIE). Notice every day a day would be a multiple of , every day a day is a multiple of , and every days it's a multiple of days without calls. Note that in the last five days of the year, days and also do not have any calls, as they are not multiples of , , or . Thus our answer is .
https://youtu.be/a3rGDEmrxC0 - Happytwin
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