Basic Math and Pre-Algebra For Dummies (30 page)

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
.

 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).

 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
  

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.