2017 AMC 8 Problems/Problem 24

Problem

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?

$\textbf{(A) }78\qquad\textbf{(B) }80\qquad\textbf{(C) }144\qquad\textbf{(D) }146\qquad\textbf{(E) }152$

Solution 1

For this problem, we use the Principle of Inclusion and Exclusion(PIE). Notice every $3$ day a day would be a multiple of $3$ , every $4$ day a day is a multiple of 4, and every 5 days it's a multiple of $144$ days without calls. Note that in the last five days of the year, days $361$ and $362$ also do not have any calls, as they are not multiples of $3$ , $4$ , or $5$ . Thus our answer is $144+2 = \boxed{\textbf{(D)}\ 146}$ .

$\[\]$

Solution 2

Since there are 365 days in the year, $365\cdot \frac45 \cdot \frac 34 \cdot \frac23=\boxed{146}$ days without calls.

~Gr8

Solution 3

We use Principle of Inclusion and Exclusion. There are $365$ days in the year, and we subtract the days that she gets at least $1$ phone call, which is $\left \lfloor \frac{365}{3} \right \rfloor + \left \lfloor \frac{365}{4} \right \rfloor + \left \lfloor \frac{365}{5} \right \rfloor$

To this result we add the number of days where she gets at least $2$ phone calls in a day because we double subtracted these days. This number is $\left \lfloor \frac{365}{12} \right \rfloor + \left \lfloor \frac{365}{15} \right \rfloor + \left \lfloor \frac{365}{20} \right \rfloor$

We now subtract the number of days where she gets three phone calls, which is $\left \lfloor \frac{365}{60} \right \rfloor$ . Therefore, our answer is

$365 - \left( \left \lfloor \frac{365}{3} \right \rfloor + \left \lfloor \frac{365}{4} \right \rfloor + \left \lfloor \frac{365}{5} \right \rfloor \right) + \left( \left \lfloor \frac{365}{12} \right \rfloor + \left \lfloor \frac{365}{15} \right \rfloor + \left \lfloor \frac{365}{20} \right \rfloor \right) - \left \lfloor \frac{365}{60} \right \rfloor$

$= 365 - 285+72 - 6 = \boxed{\textbf{(D) } 146}$ .

Solution 4

We know that the next year is 2017, and seeing that it's not a leap year, we know that there is a total of $365$ days. If x represents that last time Mrs. Sanders was called by each of her grandchildren, we can get these 3 expressions: $x+3, x+4, x+5$

Video Solution

https://youtu.be/a3rGDEmrxC0 - Happytwin

https://youtu.be/Zhsb5lv6jCI?t=2797

2017 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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