I can do this problem long hand, but was wondering if there were an easier way to do it with LCM's

Question

Earth's space colony will be floating through the area between Mars and Jupiter. There are five asteroids, named Alpha, Beta, Cephalon, Delta, and Erthon, that will occasionally orbit the colony. The scientists are advising everyone to stay inside during any time that more than one of these asteroids orbits the colony at the same time. Alpha will orbit every 6 days, Beta every 4 days, Cephalon every 9 days, Delta every 12 days, and Erthon every 18 days. How many days will the inhabitants of the space colony have to spend inside in the next 144 days?

Answer 1

Here are the first 72 days (each set of LCM=36 days are the same):

 Alpha:
_____X_____X_____X_____X_____X_____X_____X_____X_____X_____X_____X_____X
Beta:
___X___X___X___X___X___X___X___X___X___X___X___X___X___X___X___X___X___X
Cephalon:
________X________X________X________X________X________X________X________X
Delta:
___________X___________X___________X___________X___________X___________X
Erthon:
_________________X_________________X_________________X_________________X
 
The question is interesting:  In how many of the next 144 days will at least two of the asteroids be orbiting the colony at the same time? 

So, to do this longhand, you counted the number of columns with at least two x’s.  The numbers are:
12
18
24
36
48
54
60
72
84
90
96
108
120
126
132
144 
 
Now consider the prime factors of the orbit frequency numbers:
Alpha:           4  =  2 *  2
Beta :            6  =  2 *      3
Cephalon:     9  =             3 * 3
Delta:          12  =  2 * 2 * 3
Erthon:        18  =  2 *       3 * 3

Two or more:       ^    ^    ^   ^
                              1   2   3   4
Two or more concurs 4 times every 2*2*3*3=36 numbers, or 4*(144/36) = 16 times.
 
The astronauts must stay inside the colony (no EVA) for 16 of the next 144 days.

Answer 2

Hi Kathleen,

 

Congrats to David - his answer is correct.

 

He correctly shows that for each 36 days (LCM for ALL 5 orbit cycles) - there are 4 orbit cycle touch points for at least two of the orbit cycles:

    1.  at 12 days (LCM:  4,6,12)

    2.  at 18 days (LCM:  6,9) 

    3.  at 24 days (LCM:  4,6,12)

    4.  at 36 days (LCM:  4,6,9,12,18)

 

Therefore, for 144 days (36 x 4) there will be 16 orbit cycle touch points (4 x 4 = 16) for at least two of the orbit cycles.

 

God bless, Jordan.  

Answer 3

The prime factors:

A: 2, 3

B: 2²

C: 3²

D: 2², 3

E: 2, 3²

 

The LCM is 2² · 3², or 36

 

During the 144 day cycle the colonists must stay inside 144/36, or 4 days.