I recently had a student write in to ask me, “Can you explain to me the reasoning behind the Least Common Multiple? I understand that you take the prime factors from each number but I have no idea why. I think if I understood why I would be better at this technique.”
Let me see if I can make this concept more approachable for you. Think about calculating the Least Common Multiple as if you were a builder getting ready to build a house. The problem is you do not know which house you are going to build. So when you show up on the job site you need to have all of the materials for each of the possible houses. The “houses” are the numbers and the “materials” that you need are the prime factors.