nd a third chicken lays an egg every 5 days. they all lay an egg today. how many days in the next 101 days will exactly 2 chickens be laying an egg?

Question

One chicken lays an egg every 2 days, another every 3 days and a third chicken lays an egg every 5 days. They all lay an egg today. How many days in the next 101 days will exactly 2 chickens be laying an egg? 

Answer 1

Let's try to picture the future days:

1:  xExEx    ExExE    xExEx    ExExE    xExEx    ExExE

2:  xxExx    ExxEx    xExxE    xxExx    ExxEx    xExxE

3:  xxxxE    xxxxE     xxxxE    xxxxE    xxxxE    xxxxE
                   2     2        2    2        2  2          2  

   
 2 at positions:  6, 10, 12, 15, 18, 20, 24 
 
Since the LCM (Least Common Multiple) of (2,3,5) is 30, the next 30 values will have the same pattern, and the next 30 and the next 30.  So, in 101 days, there will be 3 sets of 30 days (with 7 days of 2 eggs) and a smaller set of 11 days (with 2 days of 2 eggs).  That’s a total of (3*7 + 2)=23 days of exactly2 eggs.