Question

A video game has three villains who appear on screen at different intervals. One villain appears every 5 seconds, a second villain appears every 10 seconds, and a third villain appears every 16 seconds. How much time passes between occasions when all three villains appear at the same time?

Answer 1

This question is simply asking: What is the LCM(5,10,16). We can solve this by finding the prime factors of these numbers and then only consider the largest one of them. What I mean by that is this:

$5=5\\ 10=5*2\\ 16=2^4 \implies \\ \lcm{5,10,16}=5*2^4=80$

So, the second time these villains appear together (after 0), is at 80 seconds, and we can conclude that it takes 80 econds for the villains to appear after they've last appeared.

Answer 2

We find the LCM of 5, 10 and 16.
To do so, we divide 5 and 10 by a prime factor, 5. This leaves us to 1, 2 and 16. Then, we can use a prime factor, 2 to divide 2 and 16 into 1 and 8, then it'll become 1, 1 and 8, which is the common factor.

Then we multiply everything together
5, 2 (the two prime factors we used) and also 1, 1 and 8.
5*2*1*1*8 = 80

We can conclude that the all 3 villains appear at the same time every 80 seconds.