When will all be together again?

Question

A group of 4 people are meeting for lunch in a restaurant today. One eats there every 8 days. The second eats there every 9 days. The third one eats there every 10 days and the fourth eats there every 12 days. Find the number of days from today that all 4 will first again meet there for lunch?

Answer 1

Find the LCM of these numbers: 8, 9, 10, 12

 

One way you can do this is by making lists counting by these numbers:

 

Start with making a list of the biggest one, 12, and stop when you find the first thing that 8, 9, and 10 also divide:

 

12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...

 

refine this by noting that you only get something that 10 divides every sixty:

 

60, 120, 180, 240, 300, ...

 

refine this more by noting that 8 only divides every other thing in this list:

 

120, 240, 360

 

Finally, 360 is something that 9 divides.

 

So 360 days after today, or almost a year from now, they will all be there on the same day.

 

---

 

If you can factor the numbers 8, 9, 10, and 12 into their prime factorizations, you can figure this out quicker:

 

8 = 2³

 

9 = 3²

 

10 = 2⋅5

 

12 = 2²⋅3

 

So you know the smallest multiple that all of these numbers divide has at least 3 2s, 2 3s and one 5 in it's factorization.

 

LCM(8, 9, 10, 12) = 2³⋅ 3²⋅ 5 = 360