Question 988759: Daphne is visited periodically by her three best friends: Alice, Beatrix and Claire. Alice visits every third day, Beatrix visits every forth day, and Claire visits every fifth day. All three friends visited daphne yesterday. How many days of the next 365 day period will exactly two friends visit her?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Every  days Alice and Beatrix will both visit on the same day.
That will happen  times, on days 12, 24, 36, ....360,
because 12 goes 30 times into 365:  .
Every  days Alice and Claire will both visit on the same day.
That will happen  times, on days 15, 30, 45, ....360,
because 15 goes 24 times into 365:  .
Every  days Beatrix and Claire will both visit on the same day.
That will happen  times, on days 18, 36, 54, ....360,
because 20 goes 18 times into 365:  .
Every  days all three friends will visit on the same day.
That will happen  times, on days 60, 120, 180, ....360,
because 60 goes 6 times into 365:  .
If we count all the times that two friends visit at the same time  ,
we would be counting  times the days that Alice, Beatrix and Claire will be visiting together,
so our count would be too high by  .
So the number of days that exactly two friends visit Daphne is
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