Basic Math and Pre-Algebra For Dummies (31 page)

 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
.

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
.

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
.

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
).

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.

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.