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Prime factorization exercise
We're asked, what is the prime factorization of 36? Let me get my little scratch paper out. So the prime factorization of 36. So let's start with the smallest prime number we know, and that is 2. And think about, does 2 go into 36? Well, sure, it does. 36 is 2 times 18. So we can write that down. 36 is 2 times 18. So now we have 36 as a product of a prime number, and 18 is clearly a composite number. It has factors other than 1 and 18. So let's try to factor this further. So is this divisible by 2? Sure. 18 is 2 times 9. So now 9 is a composite number that we haven't fully factored. Obviously, the 2's are both prime. 9 is not divisible by 2, but it is divisible by 3. 9 is 3 times 3. So we can say that 36 is equal to 2 times 2 times 3 times 3. This is its prime factorization. All of these numbers are prime. So now let's input that to make sure we got it right. 2 times 2 times 3 times 3. And you can check yourself. If you have the product of numbers that are all prime and the product actually is 36, you have successfully prime factorized the number. Let's do a couple more of these. What is the prime factorization of 30? So I'll get my scratch paper out again. So we'll do the same process. So 30-- well, it's divisible by 2. So we can write that as 2 times 15. 15 isn't divisible by 2. But it is divisible by 3. It's the same thing as 3 times 5. And both 3 and 5 are prime numbers. They are only divisible by 1 and themselves. So the prime factorization of 30 is 2 times 3 times 5. Let's enter that in. So it is 2 times 3 times 5. Let's do one more of these. What is the prime factorization of 73? Now, 73 is interesting. I'll get my scratch paper out for this. We could try to factor 73. So you might try 2. Well, this is clearly an odd number. So 2 isn't going to be divisible into 73. You might try 3. You would immediately see, well, 3 is divisible into 72. If you divide into 73, you have a remainder of 1. Well, 4 isn't a prime number, so we wouldn't even try. 5 isn't divisible into 73. It doesn't end in a 5 or 0. 7 is not divisible into 73. 7 goes into 70, so you'd have a remainder of 3. 11 isn't divisible into 73. It's divisible into 66 or 77, so not 73. As I test more and more numbers, it doesn't look like there's any easy thing that divides into 73. So I'm willing to go with 73 itself is a prime number. So this is its prime factorization. It's just 73. So let's write that down. So the answer here, let's just write 73. And you don't want to write 1 times 73, because 1 is not a prime number. Remember, 1 only has one factor, itself. A prime number has two factors, 1 and itself. Two different prime factors-- 1 and itself. And itself is not one. So we just want to write prime numbers here. 73 is a prime number. Let's check our answer. And we got it right.