jmc

We use Principle of Inclusion-Exclusion. There are $365$ days in the year, and we subtract the days that she gets at least $1$ phone call, which is $$\lfloor \frac{365}{3}\rfloor +\lfloor \frac{365}{4}\rfloor +\lfloor \frac{365}{5}\rfloor .$$ 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, which is $$\lfloor \frac{365}{12}\rfloor +\lfloor \frac{365}{15}\rfloor +\lfloor \frac{365}{20}\rfloor .$$ We now subtract the number of days where she gets three phone calls, which is $\lfloor \frac{365}{60}\rfloor .$ Therefore, our answer is $$365-\left(\lfloor \frac{365}{3}\rfloor +\lfloor \frac{365}{4}\rfloor +\lfloor \frac{365}{5}\rfloor \right)+\left(\lfloor \frac{365}{12}\rfloor +\lfloor \frac{365}{15}\rfloor +\lfloor \frac{365}{20}\rfloor \right)-\lfloor \frac{365}{60}\rfloor =365-285+72-6=\boxed{146}.$$