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SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

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Lesson Transcript

Instructor:
*Betty Bundly*

Betty has a master's degree in mathematics and 10 years experience teaching college mathematics.

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.

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 5x3.1 be considered a multiple? Yes, because even though 3.1 is not an integer, it is multiplied by an integer so 5x3.1 would be considered a multiple of 3.1.

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.

Now, what is prime factorization? **Prime factorization** expresses a number as the product of only prime factors and you can do this only one way for every number that can be factored. To factor a number into primes, start with any factored form, and then check each factor to see if it can be factored further. Do this until all factors are prime numbers.

For example, to factor 24 in primes,

- Start with any factored form for 24 - for example 8 x 3.
- Check both 8 and 3 to see if they are prime. 3 is prime while 8 is not. Rewrite 8 as 4 x 2.

24 = 3 x 4 x 2.

- Check these new factors to see if they are prime. 2 is prime but 4 is not. Rewrite 4 as 2 x 2.

24 = 3 x 2 x 2 x 2.

Since both 3 and 2 are prime numbers, this is the prime factorization for 24.

Now, let's use this method to find the LCM for a pair of larger numbers, 30 and 18.

- Factor both 12 and 30 into primes. 30 = 3 x 2 x 5 and 18 = 3 x 3 x 2.
- Write all the factors from the first number, then attach any factors from the second number that aren't already present.

The LCM must contain 3 x 2 x 5 from 30. Since 3 is a factor twice in 18, it must be a factor twice in the LCM. Since 2 is already listed from the factors of 30, you do not need to list it again from 18. The LCM = 3 x 2 x 5 x 3 = 90.

A **multiple** of a number is that number multiplied by an integer. Multiples of numbers are introduced in the multiplication tables. The LCM or **least common multiple** can be found for two or more numbers. The LCM is the smallest number that is a multiple of each of the numbers being considered. For small numbers, you may find the LCM by listing the multiples of the numbers. For larger numbers, the LCM can be found using the **prime factorization** method.

- A multiple of a number is that number multiplied by an integer
- A common multiple is a number that is a multiple for all the numbers being considered
- There are two ways to find the LCM:
- For small numbers, list the multiples of both numbers
- For larger numbers, identify the LCM by factoring each number into primes.

When you are finished, you should be able to:

- Explain what a multiple is
- Identify the multiples of a number
- Determine the lowest common multiple of two or more numbers

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SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

- What are the Different Types of Numbers? 6:56
- What Is The Order of Operations in Math? - Definition & Examples 5:50
- How to Find the Prime Factorization of a Number 5:36
- How to Find the Greatest Common Factor 4:56
- How to Find the Least Common Multiple 5:37
- How to Build and Reduce Fractions 3:55
- How to Find Least Common Denominators 4:30
- Comparing and Ordering Fractions 7:33
- Estimation Problems using Fractions 7:37
- How to Solve Complex Fractions 5:20
- Calculations with Ratios and Proportions 5:35
- What is a Percent? - Definition & Examples 4:20
- Changing Between Decimals and Percents 4:53
- Changing Between Decimals and Fractions 7:17
- Mathematical Sets: Elements, Intersections & Unions 3:02
- Ruler Postulate: Definition & Examples 5:19
- What is a Fact Family? - Definition & Examples 6:09
- What is a Multiple in Math? - Definition & Overview 6:02
- Basic Arithmetic: Rules & Concepts 5:02
- How to Do Double Digit Multiplication: Steps & Practice Problems 4:27
- How to Teach Double Digit Multiplication
- Double Digit Multiplication Strategies 5:10
- Counting On in Math: Definition & Strategy
- What are Turn Around Facts in Math? 3:48
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