Question

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?

$78$

$80$

$144$

$146$

$152$

Answer: D

Concepts:inclusion-exclusion least common multiple

Difficulty rating: 1800

Video solution:

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Written solution:

In a $60$ -day period, the first child calls $20$ times, the second child calls $15$ times, and the third child calls $12$ times. $20 + 15 + 12 = 47$ overcounts, however. The first and second children call on the same day $60 / 12 = 5$ times. The first and third children call on the same day $60 / 15 = 4$ times. The second and third children call on the same day $60 / 20 = 3$ times. Subtracting these from $47$ yields $47 - 5 - 4 - 3 = 35.$

The $60$ th day is added in thrice and subtracted out thrice, so we need to add it back in. This means that for every $60$ days, Mrs. Sanders receives a call $36$ days, which means that she does not receive a call on $24$ days. There are $6$ $60$ -day periods, and there are no calls on the $361$ st or $362$ nd day, which results in $24 \cdot 6 + 2 = 146$ total days with no phone calls.

Thus, D is the correct answer.

2017 AMC 8 Problem 24

Answer: D

Difficulty rating: 1800

Video solution:

Written solution:

Problem 24 in Other Years