Basic Math and Pre-Algebra For Dummies (32 page)

 
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
.

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
).

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.

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.

After you do the first step, the rest of the tree becomes easy (see Figure 
8-8
):

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.

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.