Let's find the LCM of 4 and 6. The first thing to understand is that the LCM is not the same thing as the greatest common factor (GCF). Students frequently get this mixed up. The GCF is the largest number that divides evenly into both numbers. In this case it is 2. To find the LCM, we must examine multiples of each number, and then see find the lowest multiple that each number has in common. When working with GCF, think "smaller," and when working with LCM, think "bigger."
Let's start by listing a few multiples of each number. Later you will be able to do this in your head. Multiples of 6 are 6 (itself), 12, 18, 24, 30, etc. Multiples of 4 are 4 (itself), 8, 12, 16, 20, 24, 28, 32, etc. Look to see what multiples appear on both lists. We see 12 and 24. If we kept both lists going, there would be infinitely more. For example, 36 is also a common multiple.
To find the LCM, we simply need the smallest common multiple. In this case it is 12. If we wanted to add fractions having denominators of 4 and 6, we would have to convert each fraction to have a denominator of 12, which we would call the lowest common denominator (LCD).
It is important to understand that although working with the LCM is usually best, we can also typically work with any common multiple, although it will require additional steps. Also understand that if we're struggling to find the LCM of two numbers, we can always just multiply the two numbers together to get some common multiple. It usually won't be the least, but it will be a common multiple that we can work with. For this example, it would be 24.
Let's try another example. What is the LCM of 5 and 10? Multiples of 5 are 5, 10, 15, 20, 25, 30, etc. Multiples of 10 are 10, 20, 30, 40, etc. The LCM is actually 10, which happened to be one of the original numbers. Remember that a number is always a multiple of itself. Now don't get confused. If we were asked for the GCF of these two numbers, it would be 5. Make sure you understand why.
One more example. What is the LCM of 7 and 11? Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, etc. Multiples of 11 are 11, 22, 33, 44, 55, 66, 77, 88, etc. The LCM is 77. That took a lot of work. There are two shortcuts. Both of these numbers are prime, which is discussed in another article. The LCM of two primes will always be the product of the two numbers. A simpler rule to follow is that if you're having trouble finding the LCM of two numbers, just multiply them, as described in Step 4. The result may or may not be the LCM, but it will be some common multiple that we can work with. For example, if I was asked to add 1/13 + 1/14, I wouldn't bother wasting time finding the actual LCM. I would multiply 13 times 14 to get 182, and I would use that as my LCD. There might have been a lower one, but this will work fine. We could always reduce the fraction later if necessary (discussed in another article).
Students should make certain that they are comfortable with this topic, and that they know the difference between LCM and GCF. This topic will come back again when we add and subtract fractions, and certainly in algebra in a more abstract way, so learn it now while you're still working with simple numbers out of context.