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.
Try this example (let’s use three numbers to make it more challenging):
What is the Least Common Multiple of 9, 20, and 42?
First you need to get the prime factors of each of the numbers. The prime factors of 9 are 3 * 3 the prime factors of 20 are 2 * 2 * 5 and the prime factors of 42 are 2 * 3 * 7.
Next you need to take each prime factor at the highest power. This is because you need to have all of the materials (prime factors) necessary to build any of the three houses (numbers). So your materials list is 2 * 2 * 3 * 3 * 5 * 7 or in other words 22 * 32 * 5 * 7. If you have these prime factors you can build any of the three numbers. For example, if you are asked to build the 20 you have the necessary 2*2*5.
Now you are also a very efficient builder so you do not want to bring more materials than you need. So you have to show up at the job site with the exactly the smallest load of materials with which you can build any of the houses. So that means that you do not want any extra prime factors. That is why the least common multiple on our example is 2 * 2 * 3 * 3 * 5 * 7. There is not a second 5 or another 7 because this is not needed.
You will not be asked to build more than one of the houses at any time. So even though if you list out the prime factors you will see three 2s (there are two of them in the 20 and one in the 42) and three 3s (two in the 9 and one in the 42) you do not need to bring all of these materials. You only need two 3s because you will only need to build the 9 or the 42 and not both. You only need two 2s because you will be asked to build the 20 or the 42 but not both.
I hope this helps to explain why you take each prime factor at its highest power. Understanding the reasoning behind the Least Common Multiple can help you to “build” a higher GMAT score.
David Newland has been teaching for Veritas Prep since 2006, and he won the Veritas Prep Instructor of the Year award in 2008. Students’ friends often call in asking when he will be teaching next because he really is a Veritas Prep and a GMAT rock star! Read more of his articles here.