2017 AMC 8 Problems/Problem 24

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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?

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

Solution

In $360$ days, there are \[360 \cdot \frac 23 \cdot \frac 34 \cdot \frac 45 = 144\] days without calls. Also note that in the last five days of the year, day $361$ and $362$ also do not have any calls. Thus our answer is $144+2 = \boxed{\textbf{(D)}\ 146}$.

~nukelauncher

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