Prime factorization
The fundamental theorem of arithmetic Ben Eater and Sal walk through the "Fundamental theorem of arithmetic" module: http://www.khanacademy.org/math/arithmetic/factors-multiples/e/the_fundamental_theorem_of_arithmetic
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- Sal: This is Sal, I'm here with Ben Eater.
- Ben: Hi Sal.
- Sal: And you've made this model, called the fundamental theorem of arithmetic!
- Ben: Very exciting!
- Sal: Very exciting, isn't this? This says that we can essentially get to
- any whole number greater than one
- Ben: Right.
- Sal: by taking products of prime numbers.
- Ben: That's right.
- Sal: That any number can be made by taking
- Ben: Just the right prime numbers
- Sal: by multiplying a bunch of prime numbers together.
- Ben: Yeah
- Sal: and right over here, were essential going to try to do that.
- Ben: and there's one and only
- one way to get to that number.
- Sal: There's only one way, there's not other
- kind of ways..
- Ben: There's only one set of prime numbers that will get you to any number
- Sal: Very, very interesting. So let's see, they say
- find the prime factorization of 42. So when their saying prime factorization,
- you know when your saying, since you wrote this, your saying
- factor this and all the numbers that when I multiply them together
- I get to 42. Well, the prime numbers.
- Ben: Prime numbers. And so we only give you
- prime numbers
- Sal: You only give the prime numbers, 2 through
- 13. It says use arrows to change the exponent
- on each prime number. So the exponent is how many times you're going to
- multiply that,or, and we have many videos on this,
- going into depth on exponents, but you could also, one way to visualize it
- is how many times your going to multiply one times its number.
- Ben:Right.
- Sal:For at least this case right over here, we're multiplying
- one times two 0 times, so the answer is
- one there. If you mulitply it by 2 once, you get a two there.
- and then if you do it by 2 twice, you get 2 times 2, is 4.
- So the prime factorization of 42. So the way I
- think well, lets use your hints. Ben: OK
- Sal: Sometimes, you've got these hints
- you should think about if their.. so lets see, I'd like a hint.
- so let's see what it tells us to do. We can
- use a factor tree to break 42 into its prime factorization.
- Which of the pime numbers divides into 42? so.
- Ben: A good place to start is think of any of those prime numbers
- and which ones divide... Sal: And maybe start with those at the smaller end...Ben: thats what I like to do, Yeah. ..Sal:And 2 is
- usually the easiest one to think of. This is an even number, 2 is going to didive into it.
- so lets see if thats what ya'll confirm in the hints. Oh, yes, right there,
- 2 goes in, and you can do this on paper if you want, or you should
- but 2 goes into 42 21 times, so I can just try doing it 21 times, but we're not
- done yet, because 21 is not a prime number, is not even listed here,
- keep listing the prime numbers, 21 is not prime,and so
- 2 does not go into 21 anymore, so we're kinda done with 2,
- 2's, so actually, lets see, we have 1 2 over there, but 3 does go into 21!
- Ben: It does. Sal: 3 does go into 21, and so lets see,
- so 21 can be factored into 3 times, so lets
- see if thats what your hint...Ben:which are both prime Sal: which are both prime!
- lets see if thats what the hint confirms. Yes! 3 times 7
- and then 7 is prime and your done Ben: Thats right!
- Sal: So you can say your gonna have, er, your gonna have this 2 right over here, your gonna have
- a 3, and then were gonna need a 7. 2 times 3 times
- 7 is 42. And were done! And we go right over here
- answer, check, click the.. need to change your answer, no no already did that.
- so now i just check the answer, and I'm done! Correct! Next question.
- and I got two, we used the hints so we dont get full credit.
- Lets do the next one. Lets see what shows up. Oh,we already know the prime factors of 21, we just did that!
- So we got a three, and we got a 7 in there, and there we go!
- Lets do another one. 18.
- So 2 goes into 18. Ben: It does. Sal: 2 goes into 18.
- and so, if I wanna do this in my head, 2 times 9, and I dont need to do this in my head, since I have hints,
- Ben: You have hints!
- Sal: You have hints, that essentially do that.so we can use our factor tree
- and break 18 into its prime factorization. It's 2 times 9!
- and 9 is odd, but not prime.2 doesnt go into it, 3 does.
- so we have 3 times 3, leaving us with 3..
- 3 is prime so were done factoring. So this is interesting, so we have a 2, and then were multiplying by 3 twice.
- 3 times 3 is 9, and times 2 is 18. And were done.
- we have not been able to prove the fundamental theoem of arithmatic wrong! Ben: Thats right!
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