Paul coughs every 8 seconds, karen coughs every 10. if they both cough at the same time, in how many seconds will they both cough together in?

This is also called the LCM or the Least Common Multiple! To find it you have to factor out both numbers:

      8=2*2*2
    10=2       *5   and I made space on purpose because there are no 5's in #8
LCM=2*2*2*5=40 seconds
Looking down from 8 to 10 you see that under first number 2 there is another 2, but under second, and third number 2 there is no other 2's. The first one is going to be written only once even though there are two of them and I even put it in bold so you would notice what I'm talking about.

Hope this helps and if you have any questions let me know!