Basic Math and Pre-Algebra For Dummies (29 page)

On the other hand, 4 is a composite number because it's divisible by three numbers: 1, 2, and 4. In this case, you have two ways to multiply two counting numbers and get a product of 4:

But 5 is a prime number because it's divisible only by 1 and 5. Here's the only way to multiply two counting numbers and get 5 as a product:

And 6 is a composite number because it's divisible by 1, 2, 3, and 6. Here are two ways to multiply two counting numbers and get a product of 6:

 Every counting number except 1 is either prime or composite. The reason 1 is neither is that it's divisible by only
one
number, which is 1.

Here's a list of the prime numbers that are less than 30:

• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
  

 Remember the first four prime numbers: 2, 3, 5, and 7. Every composite number less than 100 is divisible by at least one of these numbers. This fact makes it easy to test whether a number under 100 is prime: Simply test it for divisibility by 2, 3, 5, and 7. If it's divisible by any of these numbers, it's composite — if not, it's prime.

For example, suppose you want to find out whether the number 79 is prime or composite without actually doing the division. Here's how you think it out, using the tricks I show you earlier in “Knowing the Divisibility Tricks”:

• 79 is an odd number, so it isn't divisible by 2.
  
• 79 has a digital root of 7 (because 7 + 9 = 16; 1 + 6 = 7), so it isn't divisible by 3.
  
• 79 doesn't end in 5 or 0, so it isn't divisible by 5.
  
• Even though there's no trick for divisibility by 7, you know that 77 is divisible by 7. So
  leaves a remainder of 2, which tells you that 79 isn't divisible by 7.
  

Because 79 is less than 100 and isn't divisible by 2, 3, 5, or 7, you know that 79 is a prime number.

Now test whether 93 is prime or composite:

• 93 is an odd number, so it isn't divisible by 2.
  
• 93 has a digital root of 3 (because 9 + 3 = 12 and 1 + 2 = 3), so 93 is divisible by 3.
  

You don't need to look further. Because 93 is divisible by 3, you know it's composite.

Chapter 8

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.

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.

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:

This setup should remind you that 10 is a factor of 40 and that 40 is a multiple of 10.

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.

 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.