Least Common Multiple Story question

Question

Four tour buses depart the terminal on the same day. The itinerary of the first bus takes  days, for the second bus it takes  days, for the third bus it takes  days, and the fourth bus takes  days. Assuming that another tour begins the same day that the bus returns to the terminal, how many days will it take until all four buses return to the terminal on the same day?

Answer 1

he LCM (Lease Common Multiple) of a group of numbers is the lowest (least) number that is a multiple of all of them (common).

First method: List the multiples and select the lowest common (in all of the lists) value:

Multiples of 3: 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 ...

Multiples of 4: 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 ...

Multiples of 5: 5 10 15 20 25 30 35 40 45 50 55 60 ...

This method is simple, but it can take a long time.

Second method: List the prime factors and reduce duplicates to the minimum number:

Prime factors of 3: 3

Prime factors of 4: 2 * 2

Prime factors of 5: 5

The least composite number that contains each list is 2*2*3*5 = 60

Another example:

Prime factors of 8:: 2*2*2

Prime factors of 12: 2*2*3

Prime factors of 16: 2*2*2*2

LCM = 2*2*2*2*3 = 48

Notice that when you substitute the factors of a number, the remaining product is the LCM.

(2*2*2)*2*3 = 8*6 = 48

(2*2*3)*2*2 = 12*4 = 48

(2*2*2*2)*3 = 16*3 = 48