what is a prime factor
i like your funny numbers but i dont know what they mean :(
First of all, we'll only be talking about positive whole numbers, so no decimals, fractions, complex numbers, etc. This is also just going to be an introduction because this topic is more complicated/nuanced than you might expect, and I don't know/understand a lot of it
With that out of the way prime number is a number that you can't get to by multiplying other numbers: 24 isn't prime because you can write it as 6×4, but 23 is prime because the only way to get to 23 (only using multiplication) is 1×23. If you're interested, I encourage you to try finding a few prime numbers. It can be a fun exercise if you haven't tried it before. I'll put all of the ones less than 100 under the cut
A prime factor is just a prime number that divides another number. You can divide 24 by 3, and 3 is prime, so 3 is a prime factor of 24
Also, if you can get to a number by multiplying other numbers, then it's a composite number. Also also, 0 and 1 are weird so just don't worry about them <3
The thing that makes this blog work though is what's known as the fundamental theorem of arithmetic (ftoa):
Like I said, you can write 24 as 6×4, but you might notice that neither 6 nor 4 are prime. If you rewrite 6 as 2×3 and 4 as 2×2, then you can then write 24 as 2×3×2×2. Now all of the numbers are prime!
But you'll notice you can also write 24 as 8×3, then write 8 as 2×4 to get 2×4×3, then write 4 as 2×2 to get 2×2×2×3. Now that all of the numbers are prime, you'll notice that they're the same as when we started with 6×4 (three 2s and one 3)
What the ftoa says is, no matter how you break down a number like this, you'll always end with the same prime numbers at the end (in this case, three 2s and one 3)
The only thing I do differently from this process is use exponents to make it shorter. In this case, you can rewrite 2×2×2 as 2³. So, when I say 24 is 2³×3, I've "factorized" 24, and 2 and 3 are its prime factors. Of course, I don't usually do this by hand or in my head; I just give the number to a computer and it does it for me (it can be fun to do it yourself though, so sometimes I'll do it in my head for the smaller numbers)
Prime numbers and the ftoa play a pretty big role in math, and there's a ton of unsolved problems related to them, which is why you'll see people make a big deal out of them
Prime numbers under 100 and other notes under the cut:
