In this lesson we will learn about multiples, a useful concept usually introduced in grammar school multiplication tables. Understanding multiples allows you to find the least common multiple or least common denominator for two or more numbers.
Multiple of a Number Defined
When you learned your times tables in grammar school, you were learning multiples. For examples, 2, 4, 6, 8, and 10 are multiples of 2. To get these numbers, you multiplied 2 by 1, 2, 3, 4, and 5, which are integers. A multiple of a number is that number multiplied by an integer. Integers are negative as well as positive, so other multiples of 2 are -2, -4, -6, -8 and -10. Would 5×3.1 be considered a multiple? Yes, because even though 3.1 is not an integer, it is multiplied by an integer so 5×3.1 would be considered a multiple of 3.1.
What is the Least Common Multiple?
If you have ever found a common denominator for two or more fractions, you have found a common multiple. For example, if you want to add 3/8 and 5/12, you must find a common denominator. A common denominator, which is another name for common multiple, is a number that is a multiple for all the numbers being considered. For example, a common multiple for 8 and 12 is 24. This means that there is an integer times 8 that will make 24 and there is an integer times 12 that will make 24. Going through the 8 time tables, 8 x 3 = 24 and going through the 12 time tables, 12 x 2 = 24.
These are not the only common multiples for 8 and 12, however. There are countless more. For example, 72 is another common multiple because 8 x 9 = 72 and 12 x 6 = 72. The number 24, however, is special because it is the smallest or lowest or least common multiple for 8 and 12. The number 24 is called the least common multiple, abbreviated LCM, for 8 and 12.
The simplest method to find an LCM is to simply list the multiples from the time tables. For example, to find the LCM for 6, 4, and 3 I could list the multiples for all three numbers until I see the same number in all three lists.
Multiples of 6: 6, 12, 18, 24, 30
Multiples of 4: 4, 8, 12, 16, 20, 24
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24
For the multiples listed, there are two numbers that occur in both lists: 12 and 24. To find the LCM, pick the smallest number. The LCM must be 12.
The listing of the multiples method works well to find the LCM for 6, 4, and 3 because these multiples are easy to find and usually even memorized. For larger numbers this may not be such a good method.
We know that 12 = 3 x 4, 12 = 6 x 2 and 12 = 2 x 2 x 3. While all three depict 12 as a product of factors, only the last one shows 12 as a product of prime factors. A prime number is one that can be divided only by 1 and itself. A partial list of the prime numbers is 2, 3, 5, 7, 11, 13, 17, 19, and 23. This list is very partial because there is no largest prime number, which means there is no end to the list of prime numbers.
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