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The fundamental theorem of arithmetic

Video transcript
Sal: This is Sal, I'm here with Ben Eater. Ben: Hi Sal. Sal: And you've made this module, Ben: I did. Sal: 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 Ben: Any number. Sal: 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 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! (both men laughing) haven't disproved it. Sal: right, very good. well, this is very neat.