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About this Lesson
 Type: Video Tutorial
 Length: 6:54
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 74 MB
 Posted: 01/28/2009
This lesson is part of the following series:
PreAlgebra Review (31 lessons, $61.38)
In this lesson, Professor Burger introduces Prime numbers and prime factorization. A Prime number is any whole number greater than one whose only factors are one and itself, for example, 2 and 5. All whole numbers greater than one are either prime, or can be broken down into prime factors. Prime factorization is a way to represent a number as the product of its factors (thus, the prime factorization of 9 would be 3*3 given that the number 3 is prime and the product of 3 and 3 is 9). Professor Burger shows you two simple, visual ways to find a number's prime factorization: a factor tree and a ladder diagram. The factor tree pulls the numbers out, creating a diagram that looks like a family tree. The ladder diagram starts with dividing the number by prime factors until you get to 1, creating a stairstep diagram as you go.
Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Pre Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/prealgebra. The full course covers whole numbers, integers, fractions and decimals, variables, expressions, equations and a variety of other pre algebra topics.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UTAustin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, padic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
About this Author
 Thinkwell
 2174 lessons
 Joined:
11/14/2008
Founded in 1997, Thinkwell has succeeded in creating "nextgeneration" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technologybased textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.
Thinkwell lessons feature a starstudded cast of outstanding university professors: Edward Burger (PreAlgebra through...
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PRIME FACTORIZATION
Well, we have already seen that the prime numbers are those whole numbers that can't be factored into two smaller whole numbers.
Now, we are going to see that in some sense the prime numbers are the basic multiplicative building blocks for all whole numbers bigger than 1.
What do I mean? What I mean is if you look at the whole number bigger than 1, it has two possibilities. It is either prime or it can be expressed as a product of prime numbers. We call that expansion into a product of prime numbers the prime factorization of the number. Now, there are a couple of ways of actually figuring out the prime factorization for a whole number, and I want to show you both of them to get a sense of how they work.
So, let's take an example, say 100, and I want to write it and express it in its prime factorization. So, what do I do? I am going to first illustrate this with what is called the factor tree. We are actually trying to develop a tree with roots. The way we are going to do it is very simple. We are going to slowly rip off prime factors, and it is a “divide and conquer” method. In fact, both of these methods that I am going to show you are divide and conquer methods.
So, look at 100 and say, "Can I write 100 as a prime times something else?" Sure, I can write it as a prime, say 2 times 50. Well, you see I have now ripped off one of the primes, and now I have a small number. Divide and conquer.
Now, I am going to see if I can write 50 as a prime times something else. Sure, I can write it as 2 times 25 and now, I take 25 and see if I can write it as a prime times something else. 5 times 5 and notice that this cannot be factored into any further
because in fact 5 is known to be a prime. So, what do I see? What I see here, 2, 2, 5,5, and so, I see that the prime factorization for 100 is 2 multiplied by 2, multiplied by 5, multiplied by 5. You can check that if you take the product.
Now, there is not just one way of producing one of these factor trees. Let me just show you a different version of a factor tree just by pulling out a different factor from the beginning.
Start with 100. Now, I am going to pull out the 5. 5 times what equals 100? Well, if you think about it, it is 20. Now, I pull out another 5 from here. I see I am left with 4 and that 4 can be written as 2 times 2. I guess if you want to go even further
you can write it as 2 times 1 and then, stop there. You can see now 5, 5, 2, 2. So, you can see the exact same factors. 100 equals 5 times 5 times 2 times 2. Notice, we get the exact same answer. The only difference is that the factors have been reordered. And that is completely ok because remember, it does not make a difference how we multiply things, any order is ok. So, these are the exact same answers. We factored this into 100 into prime factorization as 2 times 2 times 5 times 5 in any order that you want. We did that using a factor tree.
Let me now show now a different method. This method is called the ladder diagram. So, let's try to illustrate that by factoring the number 72. So, I am going to use 72 as an example to show how to factor numbers using this ladder diagram.
What you do, is, you start with 72 and you think of a factor of 72. I see it is even so I know 2 should divide that. So, I start doing an upside down division sign with the 2 up there and then divide the 2 into 72 and I see 36. Now, I find a other factor of 36. I pick 2 and I divide 36, take 2 into it, I see 18. Try to factor here. Well, about 2 again? Then, I am left with a 9. I do the division. I think of a factor there. How about 3? That leaves me with 3. I see that 3 goes into it and I get to 1. When we get down to 1, that means the game is over. That means we stop right there. The prime factorization is these numbers right here: 2 times 2 times 2 times 3 times 3. It is going to equal to 72. That's the beauty of the ladder diagram. You can just read off the prime factorization by just pulling off these primes one at the time. But just like with the tree diagram, there are a lot of ways to go. I don't want you to think, "Oh, there is a only one way of doing the ladder diagram.” There are a lot of different ways. Let me show this one another way.
Suppose, you get 72, took 7 and add 2, you got 9. Realize that 3 goes into 9 which means that 3 is a factor of 72. So, let's start with 3. So, if we start with 3 then when I do the division, I see 24. Now I notice 24, that's even. So, I pull out a 2, do
the division, and I get 12. There is another 3 in there, I pull out the 3 and I see a 4. That's even. I can factor out 2, get a 2; factor a 2 again, and I get a 1. Game is over. So, now, it is a completely different ladder diagram but look at what's on the
ladder! 3, 2, 3, 2, 2. Maybe it sounds different but look, it is the exact same thing here. Here, I see three 2s and two 3s. Here, I see three 2s and two 3s.
It is the same factorization, just written in different order. So, it turns out that the prime factorization of any number is absolutely unique, except for the order that we write the factors in. You can find those factors, either by using the ladder diagram or using the tree figure. So, both would work and both lead to the prime factorization for any whole number you can think off, that's bigger than 1. If you can't factor, if you immediately go down to 1, that means that the number is prime. Have fun with the primes. They are really a wonderful object!
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Excellent job! This really made everything clear.