2017 AMC 8 Problem 24

Calls repeat every $\operatorname{lcm}(3,4,5)$ days.

Count days with at least one call in each $60$-day cycle, then handle the last few days of $2017.$

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 $\frac{60}{12} = 5$ times. The first and third children call on the same day $\frac{60}{15} = 4$ times. The second and third children call on the same day $\frac{60}{20} = 3$ 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.