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How To Find The Least Common Multiple (LCM) Of Two Numbers

 
A common task in math is to find the least common multiple (or LCM as abbreviated by Abbreviation Finder) of two numbers. If we do this for the sake of adding or subtracting fractions with unlike denominators (discussed in another article), then this number will be used as the lowest common denominator (or LCD as abbreviated by Abbreviation Finder) for the two factions.

Note that there are many clever methods to calculate the LCM of two numbers. Most of these are not at all practical, since they take a very long time, and the risk of error. They are typically only studied once, when the subject will be first be taught, and then never again. In later math and on standardized exams, we are never given large and obscure figures to work with. It is never the point. This article shows you the steps to quickly find the LCM of two rational numbers, maybe even in your head.

1. Let us find the LCM of 4 and 6. The first thing to understand is that the LCM is not the same as the greatest common factor (or GCF as abbreviated by Abbreviation Finder). Students often 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 see then find the lowest multiple that each number has in common. When you are working with GCF, think "less" and when working with LCM, think "bigger".

2. 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 (even), 12, 18, 24, 30, etc. multiples of 4 are 4 (even), 8, 12, 16, 20, 24, 28, 32, etc. Look to see what multiples appear on both lists. We see 12 and 24 years. If we kept both lists going, there would be infinitely more. For example 36 is also a common multiple.

3. To find the LCM, we simply need the least common multiple. In this case, it's 12. If we wanted to add fractions with denominators of 4 and 6, we would have to convert each fraction has a denominator of 12, which we would call the lowest common denominator (LCD).

4. It is important to understand that although the work of LCM is usually best, can we also typically work with any common multiple, although it will require additional steps. Also understand that if we are struggling to find the LCM of two numbers, we can always just multiply the two numbers together to get some common multiple. It will usually not be the smallest, but it will be a common multiple that we can work with. For this example, it would be 24.

5. 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 figures. Keep in mind that a number is always a multiple of itself. Now don't get confused. If we were asked the GCF of these two numbers, it would be 5. Make sure you understand why.

6. Yet another 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. are multiples of 11 are 11, 22, 33, 44, 55, 66, 77, 88, etc. The LCM is 77. It took a lot of work. There are two shortcuts. Both of these numbers is 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 have 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 on finding the actual LCM. I would multiply 13 times 14 to get 182, and I want to use it as my LCD. There may have been a lower, but it will work just fine. We can always reduce the fraction later if necessary (discussed in another article).

7. Students should ensure 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 put together 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.
 
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