Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)
Part II. Getting a Handle on Whole Numbers
Chapter 8. Fabulous Factors and Marvelous Multiples
In This Chapter
Understanding how factors and multiples are related
Listing all the factors of a number
Breaking down a number into its prime factors
Generating multiples of a number
Finding the greatest common factor (GCF) and least common multiple (LCM)
In Chapter 2, I introduce you to sequences of numbers based on the multiplication table. In this chapter, I tell you about two important ways to think about these sequences: as factors and as multiples. Factors and multiples are really two sides of the same coin. Here I show you what you need to know about these two important concepts.
For starters, I discuss how factors and multiples are connected to multiplication and division. Then I show you how to find all the factor pairs of a number and how to decompose (split up) any number into its prime factors. To finish up on factors, I show you how to find the greatest common factor (GCF) of any set of numbers. After that, I tackle multiples, showing you how to generate the multiples of a number and then use this skill to find the least common multiple (LCM) of a set of numbers.
Knowing Six Ways to Say the Same Thing
In this section, I introduce you to factors and multiples, and I show you how these two important concepts are connected. As I discuss in Chapter 4, multiplication and division are inverse operations. For example, the following equation is true:
So this equation using the inverse operation is also true:
You may have noticed that, in math, you tend to run into the same ideas over and over again. For example, mathematicians have six different ways to talk about this relationship.
The following three statements all focus on the relationship between 5 and 20 from the perspective of multiplication:
· 5 multiplied by some number is 20.
· 5 is a factor of 20.
· 20 is a multiple of 5.
In two of the examples, you can see this relationship reflected in the words multiplied and multiple. For the remaining example, keep in mind that two factors are multiplied to equal a product.
Similarly, the following three statements all focus on the relationship between 5 and 20 from the perspective of division:
· 20 divided by some number is 5.
· 20 is divisible by 5.
· 5 is a divisor of 20.
Why do mathematicians need all these words for the same thing? Maybe for the same reason that Eskimos need a bunch of words for snow. In any case, in this chapter, I focus on the words factor and multiple. When you understand the concepts, which word you choose doesn't matter a whole lot.
Connecting Factors and Multiples
When one number is a factor of a second number, the second number is a multiple of the first number. For example, 20 is divisible by 5, so
· 5 is a factor of 20.
· 20 is a multiple of 5.
Don't mix which number is the factor and which is the multiple. The factor is always the smaller number, and the multiple is always the larger number for positive numbers.
If you have trouble remembering which number is the factor and which is the multiple, jot them down in order from lowest to highest, and write the letters F and M in alphabetical order under them.
For example, 10 divides 40 evenly, so jot down:
10 |
40 |
F |
M |
This setup should remind you that 10 is a factor of 40 and that 40 is a multiple of 10.
Finding Fabulous Factors
In this section, I introduce you to factors. First, I show you how to find out whether one number is a factor of another. Then I show you how to list all the factor pairs of a number. After that, I introduce the key idea of a number's prime factors. This information all leads up to an essential skill: finding the greatest common factor (GCF) of a set of numbers.
Deciding when one number is a factor of another
You can easily tell whether a number is a factor of a second number: Just divide the second number by the first. If it divides evenly (with no remainder), the number is a factor; otherwise, it's not a factor.
For example, suppose you want to know whether 7 is a factor of 56. Here's how you find out:
Because 7 divides 56 without leaving a remainder, 7 is a factor of 56.
And here's how you find out whether 4 is a factor of 34:
Because 4 divides 34 with a remainder of 2, 4 isn't a factor of 34.
This method works no matter how large the numbers are.
Some teachers use factoring problems to test you on long division. For a refresher on how to do long division, see Chapter 3.
Understanding factor pairs
A factor pair of a number is any pair of two numbers that, when multiplied together, equal that number. For example, 35 has two factor pairs — 1 × 35 and 5 × 7 because
Similarly, 24 has four factor pairs — — because
Every positive integer has at least one factor pair: 1 times the number itself. For example:
When a number greater than 1 has only one factor pair, it's a prime number (see Chapter 7 for more on prime numbers).
Generating a number's factors
The greatest factor of any number is the number itself, so you can always list all the factors of any number because you have a stopping point. A good way to list all the factors of a number is to list all its factor pairs:
1. Begin the list with 1 times the number itself.
2. Try to find a factor pair that includes 2.
That is, see whether the number is divisible by 2 (for tricks on testing for divisibility, see Chapter 7). If it is, list the factor pair that includes 2.
3. Test the number 3 in the same way.
4. Continue testing numbers until you find no more factor pairs.
An example can help make this clear. Suppose you want to list all the factors of the number 18. According to Step 1, begin with 1 × 18:
· 1 × 18
Remember from Chapter 7 that every number — whether prime or composite — is divisible by itself and 1. So automatically, 1 and 18 are both factors of 18.
Next, see if you can find a factor pair of 18 that includes 2. Of course, 18 is an even number, so you know that such a factor pair exists. (For a bunch of easy divisibility tricks, check out Chapter 3.) Here it is:
· 2 × 9
Because 2 divides 18 without a remainder, 2 is a factor of 18. (For a bunch of easy divisibility tricks, check out Chapter 3.) So both 2 and 9 are factors of 18, and you can add them both to the list:
Now test 3 in the same way:
· 3 × 6
At this point, you're almost done. You have to check only the numbers between 3 and 6 — that is, the numbers 4 and 5. Neither of these numbers is included in a factor pair of 18 because 18 isn't divisible by 4 or 5:
So 18 has three factor pairs — 1 × 18, 2 × 9, and 3 × 6 — and thus has six factors. If you like (or if your teacher prefers!), you can list these factors in order, as follows:
· 1 2 3 6 9 18
Identifying prime factors
In Chapter 7, I discuss prime numbers and composite numbers. A prime number is divisible only by 1 and itself — for example, the number 7 is divisible only by 1 and 7. On the other hand, a composite number is divisible by at least one number other than 1 and itself — for example, the number 9 is divisible not only by 1 and 9, but also by 3.
A number's prime factors are the set of prime numbers (including repeats) that equal that number when multiplied together. For example, here are the prime factors of the numbers 10, 30, and 72:
In the last example, the prime factors of 72 include the number 2 repeated three times and the number 3 repeated twice.
The best way to break down a composite number into its prime factors is to use a factorization tree. Here's how it works:
1. Split the number into any two factors and check off the original number.
2. If either of these factors is prime, circle it.
3. Repeat Steps 1 and 2 for any number that is neither circled nor checked.
4. When every number in the tree is either checked or circled, the tree is finished, and the circled numbers are the prime factors of the original number.
For example, to break down the number 56 into its prime factors, start by finding two numbers (other than 1 or 56) that, when multiplied, give you a product of 56. In this case, remember that 7 × 8 = 56. See Figure 8-1.
Illustration by Wiley, Composition Services Graphics
Figure 8-1: Finding two factors of 56; 7 is prime.
As you can see, I break down 56 into two factors and check it off. I also circle 7 because it's a prime number. Now, 8 is a neither checked nor circled, so I repeat the process, as shown in Figure 8-2.
Illustration by Wiley, Composition Services Graphics
Figure 8-2: Continuing the number breakdown with 8.
This time, I break 8 into two factors (2 × 4 = 8) and check it off. This time, 2 is prime, so I circle it. But 4 remains unchecked and uncircled, so I continue with Figure 8-3.
Illustration by Wiley, Composition Services Graphics
Figure 8-3: The finished tree, completed from Figure 8-1.
At this point, every number in the tree is either circled or checked, so the tree is finished. The four circled numbers — 2, 2, 2, and 7 — are the prime factors of 56. To check this result, just multiply the prime factors:
You can see why this is called a tree: Starting at the top, the numbers tend to branch off like an upside-down tree.
What happens when you try to build a tree starting with a prime number — for example, 7? Well, you don't have to go very far (see Figure 8-4).
Illustration by Wiley, Composition Services Graphics
Figure 8-4: Starting with a prime number.
That's it — you're done! This example shows you that every prime number is its own prime factor.
Here's a list of numbers less than 20 with their prime factorizations. (As you find out in Chapter 2, 1 is neither prime nor composite, so it doesn't have a prime factorization.)
As you can see, the eight prime numbers that I list here are their own prime factorizations. The remaining numbers are composite, so they can all be broken down into smaller prime factors.
Every number has a unique prime factorization. This fact is important — so important that it's called the Fundamental Theorem of Arithmetic. In a way, a number's prime factorization is like its fingerprint — a unique and foolproof way to identify a number.
Knowing how to break down a number to its prime factorization is a handy skill to have. Using the factorization tree allows you to factor out one number after another until all you're left with are primes.
Finding prime factorizations for numbers 100 or less
When you build a factorization tree, the first step is usually the hardest. That's because, as you proceed, the numbers get smaller and easier to work with. With fairly small numbers, the factorization tree is usually easy to use.
As the number you're trying to factor increases, you may find the first step to be a little more difficult. It's especially hard when you don't recognize the number from the multiplication table. The trick is to find someplace to start.
Whenever possible, factor out 5s and 2s first. As I discuss in Chapter 7, you can easily tell when a number is divisible by 2 or by 5.
For example, suppose you want the prime factorization of the number 84. Because you know that 84 is divisible by 2, you can factor out a 2, as shown in Figure 8-5.
Illustration by Wiley, Composition Services Graphics
Figure 8-5: Factoring out 2 from 84.
At this point, you should recognize 42 from the multiplication table (6 × 7 = 42).
This tree is now easy to complete (see Figure 8-6).
Illustration by Wiley, Composition Services Graphics
Figure 8-6: Completing the factoring of 84.
The resulting prime factorization for 84 is as follows:
If you like, though, you can rearrange the factors from lowest to highest:
By far, the most difficult situation occurs when you're trying to find the prime factors of a prime number but don't know it. For example, suppose you want to find the prime factorization for the number 71. This time, you don't recognize the number from the multiplication tables, and it isn't divisible by 2 or 5. What next?
If a number that's less than 100 (actually, less than 121) isn't divisible by 2, 3, 5, or 7, it's a prime number.
Testing for divisibility by 3 by finding the digital root of 71 (that is, by adding the digits) is easy. As I explain in Chapter 7, numbers divisible by 3 have digital roots of 3, 6, or 9.
· 7 + 1 = 8
Because the digital root of 71 is 8, 71 isn't divisible by 3. Divide to test whether 71 is divisible by 7:
· 71 ÷ 7 = 10r1
So now you know that 71 isn't divisible by 2, 3, 5, or 7. Therefore, 71 is a prime number, so you're done.
Finding prime factorizations for numbers greater than 100
Most of the time, you don't have to worry about finding the prime factorizations of numbers greater than 100. Just in case, though, here's what you need to know.
As I mention in the preceding section, factor out the 5s and 2s first. A special case is when the number you're factoring ends in one or more 0s. In this case, you can factor out a 10 for every 0. For example, Figure 8-7 shows the first step.
Illustration by Wiley, Composition Services Graphics
Figure 8-7: The first step in factoring 700.
After you do the first step, the rest of the tree becomes easy (see Figure 8-8):
Illustration by Wiley, Composition Services Graphics
Figure 8-8: Completing the factoring of 700.
You can see that the prime factorization of 700 is
If the number isn't divisible by either 2 or 5, use your divisibility trick for 3 (see Chapter 7) and factor out as many 3s as you can. Then factor out 7s, if possible (sorry, I don't have a trick for 7s), and, finally, 11s.
If a number that's less than 289 isn't divisible by 2, 3, 5, 7, 11, or 13, it's prime. As always, every prime number is its own prime factorization, so when you know that a number is prime, you're done. Most of the time, with larger numbers, a combination of tricks can handle the job.
Finding the greatest common factor (GCF)
When you understand how to find the factors of a number (see “Generating a number's factors”), you're ready to move on to the main event: finding the greatest common factor of several numbers.
The greatest common factor (GCF) of a set of numbers is the largest number that's a factor of all those numbers. For example, the GCF of the numbers 4 and 6 is 2 because 2 is the greatest number that's a factor of both 4 and 6.
To find the GCF of a set of numbers, list all the factors of each number, as I show you in “Generating a number's factors.” The greatest factor appearing on every list is the GCF. For example, to find the GCF of 6 and 15, first list all the factors of each number.
· Factors of 6: 1, 2, 3, 6
· Factors of 15: 1, 3, 5, 15
Because 3 is the greatest factor that appears on both lists, 3 is the GCF of 6 and 15.
As another example, suppose you want to find the GCF of 9, 20, and 25. Start by listing the factors of each:
· Factors of 9: 1, 3, 9
· Factors of 20: 1, 2, 4, 5, 10, 20
· Factors of 25: 1, 5, 25
In this case, the only factor that appears on all three lists is 1, so 1 is the GCF of 9, 20, and 25.
Making Marvelous Multiples
Even though multiples tend to be larger numbers than factors, most students find them easier to work with. Read on.
Generating multiples
The preceding section, “Finding Fabulous Factors,” tells you how to find all the factors of a number. Finding all the factors is possible because factors of a number are always less than or equal to the number itself. So no matter how large a number is, it always has a finite (limited) number of factors.
Unlike factors, multiples of a number are greater than or equal to the number itself. (The only exception to this is 0, which is a multiple of every number.) Because of this, the multiples of a number go on forever — that is, they're infinite. Nevertheless, generating a partial list of multiples for any number is simple.
To list multiples of any number, write down that number and then multiply it by 2, 3, 4, and so forth.
For example, here are the first few positive multiples of 7:
· 7 14 21 28 35 42
As you can see, this list of multiples is simply part of the multiplication table for the number 7. (For the multiplication table up to 9 × 9, see Chapter 3.)
Finding the least common multiple (LCM)
The least common multiple (LCM) of a set of numbers is the lowest positive number that's a multiple of every number in that set.
For example, the LCM of the numbers 2, 3, and 5 is 30 because
· 30 is a multiple of 2 (2 × 15 = 30).
· 30 is a multiple of 3 (3 × 10 = 30).
· 30 is a multiple of 5 (5 × 6 = 30).
· No number lower than 30 is a multiple of all three numbers.
To find the LCM of a set of numbers, take each number in the set and jot down a list of the first several multiples in order. The LCM is the first number that appears on every list.
When looking for the LCM of two numbers, start by listing multiples of the larger number, but stop this list when the number of multiples you've written equals the smaller number. Then start listing multiples of the lower number until one of them matches the first list.
For example, suppose you want to find the LCM of 4 and 6. Begin by listing multiples of the higher number, which is 6. In this case, list only four of these multiples because the lower number is 4.
·
Multiples of 6: |
6, 12, 18, 24, … |
Now start listing multiples of 4:
·
Multiples of 4: |
4, 8, 12, … |
Because 12 is the first number to appear on both lists of multiples, 12 is the LCM of 4 and 6.
This method works especially well when you want to find the LCM of two numbers, but it may take longer if you have more numbers.
Suppose, for instance, that you want to find the LCM of 2, 3, and 5. Start with the two largest numbers — in this case, 5 and 3 — and keep listing numbers until you have one or more matching numbers:
·
Multiples of 5: |
5, 10, 15, 20, 25, 30, … |
Multiples of 3: |
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, … |
The only numbers repeated on both lists are 15 and 30. In this case, you can save yourself the trouble of making the last list because 30 is obviously a multiple of 2, and 15 isn't. So 30 is the LCM of 2, 3, and 5.