Factoring

In order to move into fractions we first have to stop and learn to factor whole numbers.

Along with learning to factor whole numbers comes understanding prime numbers, least common multiples and greatest common factors. All of this is explained below.

After completing this section move on to fractions.

Factoring Whole Numbers

Factoring whole numbers is easy to learn if you know how to add and multiply WELL

Tip #1

In this next section, we’re going to learn to factor whole numbers. This will help us reduce, add and subtract fractions. First, let’s find out what factoring is and then learn how to factor.

The two or more numbers that can be multiplied together to get another given number are its factors. So to factor the number 9 for example, we know that 3 × 3 = 9, so 3 would be a factor of 9. We also know that 1 × 9 = 9 so 1 and 9 are factors of 9, too. So 1, 3 and 9 are all factors of 9.

To factor 6, we’d think about the numbers that can be multiplied together to get 6. We know 2 x 3 = 6, and 1 × 6 = 6….so 1, 2, 3 and 6 all factors of 6.

To factor any number, we need to determine all the numbers that can be multiplied together to get the other given number. This is very easy to do if you are a multiplication champion and learn about Prime Numbers, which can’t be factored, and follow below.

In another example, 10 can be factored by the numbers 1, 2, 5, and 10.
(again, the factors 1, 2, 5, and 10 come from: (1 × 10) and (2 × 5) since these numbers can be multiplied in some way to equal 10).

There are three tools to help us factor. One of the tools is to know the multiplication tables. The other tools for factoring are to know the multiples of the numbers 1 – 10 and how to divide. Knowing the times tables, multiples and division will help us breakout the factors of numbers 4 – 100.

For example, the multiples of 2 are (2, 4, 6, 8, 10, 12, 14, 16, 18, 20).
And the multiples of 3 are (3, 6, 9, 12, 15, 18, 21, 24, 27, 30).
Now, let’s say we’re asked to factor 18. We can scan our multiples and see that 2 would be one of the factors since 9 × 2 = 18. We can also scan the multiples of 3 and see that 3 would be one of the factors since 6 × 3 = 18. So knowing these multiples can really help.

Let’s try to factor 6 using the multiples listed in the last problem.

Are 2 and 3 factors of 6?

Yep, the numbers 1, 2, 3, and 6 are all factors of 6.
(again, the factors 1, 2, 3, and 6 come from: 1 X 6 = 6 and 2 X 3 = 6).

OK, here’s your chance to see if you understand factoring. Use the area below to factor the following numbers. If you have trouble, it may help to have a multiplication table handy.

Sometimes we run into larger numbers that need to be factored. For example, the number 120 has quite a few factors…1, 2, 3, 4, 6, 8, 10, 12, 20, 30, 40, 60, 120. One rule when factoring: remember to list ALL the factors of the number!

Some suggestions for factoring numbers is to first, try to divide the number in half or by 2. Then, if you’re trying to factor a number and it’s a number that’s divisible by 3 (meaning 3, 6, 9, 12 or as the last digits….63 for example) use multiples of 3’s to factor it (3, 6, 9,…). If a number ends in 5, (like the number 155 for example) try factoring with multiples of 5 (5, 10, 15,…etc). With a little practice, factoring is no more difficult than division.

And finally, to make sure you get all the factors of a number it really helps to use a calculator.

Prime Numbers

Prime numbers are whole numbers greater than one and are divisible only by one and itself. In other words, Prime Numbers Can’t Be Factored by numbers other than 1 and itself! The numbers 2, 3, 5, 7, 11, 13, 17, 23, 37,…are all prime numbers. Two is the only even prime number, all others are odd.

To show what a prime isn’t, take a number like 4. It’s between two prime numbers, 3 and 5.
What makes 4 not prime is that 2 x 2 and 1 x 4 both equal 4 so they both factor it. If it were prime only 1 × 4 would be it’s factor. So, 4 can be factored by 1, 2 and 4. 4 is called a COMPOSITE number. Composite means more than one set of factors are used to make the number. Other examples of composite numbers are 6, 8, 10, 12, etc… as you can see, each of these composite numbers can be factored by numbers other than 1 and the number.

Remember, a prime number can only be factored by 1 and itself!!!

The prime number 3 can only be factored by multiplying 1 x 3. It can’t be factored by another set of whole numbers. Sure, we could factor numbers like 3 but we end up with partial whole numbers. For example, say we try to factor 3 by dividing it by 2. We end up with 1.5 X 2 = 3. Since 1.5 is not a whole number we say that 3 can’t be factored evenly by 2. When factoring correctly, we will always end up with whole numbers as a result.

Same with 5.
A 5 can only be factored evenly by 1 x 5 = 5 that’s why 5 is a prime number.

Greatest Common Factor

The reason we want to know about prime numbers is to help us factor. One way to factor is to find the largest number shared by both numbers. This is called the greatest common factor. Say we want to factor two numbers, for example, the numbers 12 and 18.

As you can see, the greatest common factor of 12 and 18 is 6. To check again…just multiply 6 × 2 = 12 and 6 × 3 = 18 that makes 6 the largest factor shared by both 12 and 18.

Here’s a little practice to find the greatest common factor.
In the area below, list all the factors of each number, then circle the largest number shared by both 16 and 28.
16 = ____ ____ ____ ____ ____ ____
28 = ____ ____ ____ ____ ____ ____

For 16 did you get 1, 2, 4, 8 and 16?.
For 28 did you get 1, 2, 4, 7, 14 and 28?

Now, look at both sets of numbers and determine the largest number shared by both 16 and 28.
Did you get 4? Yep, 4 is the greatest common factor shared by 16 and 28….easy isn’t it?

Maybe for some of you that wasn’t so easy….so here’s another way to find greatest common factors.

List all of the prime factors who’s product equals the numbers 16 and 28.

Keep in mind that prime factors don’t include 1 or the number (either 16 or 28 in this case). So let’s list those here

16 = 2 × 2 × 2 × 2
28 = 2 × 2 × 7

Now multiply the common prime factors (2 × 2) and the product equals 4 which is the greatest common factor of 16 and 28.

(Place new activity here for greatest common factors)

Least Common Multiple

Finding the least common multiple is an important skill for further dealings with fractions. We will be seeking the least common multiples for adding and subtracting fractions. So be extra careful and understand this section completely before moving onto other areas of fractions.

The least common multiple is the smallest whole number that is divisible by two or more numbers. For example, let’s take two numbers, say 16 and 34. Then let’s multiply 1, 2, 3, 4, etc., to each of the numbers. When we hit the two smallest numbers shared by both numbers we have found the least common multiple.

16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272, 288
34 = 34, 68, 102, 136, 170, 204, 238, 272, 306

As you can see, the least common multiple of 16 and 34 is 272. This is really not difficult, but it is time consuming. So when finding the least common multiple of two numbers remember to relax and take your time….this isn’t a race.

Another way to find the least common multiple is to prime factor both numbers like this:
16 = 2 × 2 × 2 × 2
34 = 2 × 17

Now count the number of times the prime numbers are in each factorization. Then, take the largest of these two counts and write that prime number down as many times as it appears.

So for 16 we find that 2 × 2 × 2 × 2 = 16
And for 34 we find that 2 × 17 = 34.

Since there are four 2s in 2 × 2 × 2 × 2 and only one 2 in factoring 34 we’ll use 2 × 2 × 2 × 2. And since 17 appears once, we’ll use that once. Now, multiply the four 2s and the one 17. The results are shown below.

2 × 2 × 2 × 2 × 17 = 272

And, as we saw in the last problem, 272 is the least common multiple of 16 and 34.