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## Pre-algebra

# Prime numbers

CCSS.Math:

Prime numbers are numbers that have only 2 factors: 1 and themselves. For example, the first 5 prime numbers are 2, 3, 5, 7, and 11. By contrast, numbers with more than 2 factors are call composite numbers. Created by Sal Khan.

## Want to join the conversation?

- What is the best way to figure out if a number (especially a large number) is prime?(440 votes)
- What I try to do is take it step by step by eliminating those that are not primes.

step 1. except number 2, all other even numbers are not primes.

step 2. except number 5, all other numbers divisible by 5 are not primes

so far so good :), now comes the harder part especially with larger numbers

step 3: I start with the next lowest prime next to number 2, which is number 3 and use long division to see if I can divide the number.

step 4: I repeat step 3 with trying to divide by the prime 5, 7 , 11 and onward until the prime is found or not ^^

hope it helps.(264 votes)

- What is the harm in considering 1 a prime number? It is true that it is divisible by itself and that it is divisible by 1, why is the "exactly 2" rule so important?(68 votes)
- The Fundamental Theorem of Arithmetic states that every number is either prime or is the product of a list of prime numbers, and that list is unique aside from the order the terms appear in. For instance, I might say that 24 = 3 x 2 x 2 x 2 and you might say 24 = 2 x 2 x 3 x 2, but we each came up with three 2's and one 3 and nobody else could do differently. This wouldn't be true if we considered 1 to be a prime number, because then someone else could say 24 = 3 x 2 x 2 x 2 x 1 and someone else could say 24 = 3 x 2 x 2 x 2 x 1 x 1 x 1 x 1 and so on, Sure, we could declare that 1 is a prime and then write an exception into the Fundamental Theorem of Arithmetic, but all in all it's less hassle to just say that 1 is neither prime nor composite.(114 votes)

- why is 1 not prime?(70 votes)
- In the 19th century some mathematicians did consider 1 to be prime, but mathemeticians have found that it causes many problems in mathematics, if you consider 1 to be prime. The most notable problem is The Fundamental Theorem of Arithmetic, which says any number greater than 1 has a unique prime factorization. e.g. 6= 2* 3, (2 and 3 being prime). But if we let 1 be prime we could write it as 6=1*2*3 or 6= 1*2 *1 *3. There would be an infinite number of ways we could write it. There are other issues, but this is probably the most well known issue.

So 1 is specifically excluded from being prime.(100 votes)

- Ate there any easy tricks to find prime numbers?(124 votes)
- If you want an actual equation, the answer to your question is much more complex than the trouble is worth. There's an equation called the Riemann Zeta Function that is defined as The Infinite Series of the summation of 1/(n^s), where "s" is a complex variable (defined as a+bi). When the "a" part, or real part, of "s" is equal to 1/2, there arises a common problem in number theory, called the Riemann Hypothesis, which says that all of the non-trivial zeroes of the function lie on that real line 1/2. The distribution of the values directly relate to the amount of primes that there are beneath the value "n" in the function. This delves into complex analysis, in which there are graphs with four dimensions, where the fourth dimension is represented by the darkness of the color of the 3-D graph at its separate values. Like I said, not a very convenient method, but interesting none-the-less.(82 votes)

- Why is 2 considered a prime number?(47 votes)
- As Sal says at0:58, it's a number divisible by only two natural numbers. And two is divisible by only 1 and 2. It does, as he mentions at2:51, break the pattern of all the other primes in being the only even prime number.(53 votes)

- Is pi prime or composite?(13 votes)
- Neither - those terms only apply to integers (whole numbers) and pi is an irrational decimal number.(71 votes)

- Is a "negative" number not natural??(27 votes)
- yes. natural ones are whole and not fractions and negatives.(5 votes)

- Why does a prime number have to be divisible by two natural numbers? Why can't it also be divisible by decimals?(34 votes)
- All numbers are divisible by decimals. For example, you can divide 7 by 2 and get 3.5 . When using prime numbers and composite numbers, stick to whole numbers, because if you are factoring out a number like 9, you wouldn't say its prime factorization is 2 x 4.5, you'd say it was 3 x 3, because there is an endless number of decimals you could use to get a whole number.(4 votes)

- Why is one not a prime number i don't understand?(7 votes)
- I have question for you

Would consider 1 and 1 different numbers?

I wouldn’t. The mathematical community doesn’t either.

So when a number is divisible by itself and one, it’s prime asking the number itself isn’t one

I hope this helps you(6 votes)

- At2:08what does counter intuitive mean ?(9 votes)
- It means that something is opposite of common-sense expectations but still true.Hope that helps! :)(8 votes)

## Video transcript

In this video, I want
to talk a little bit about what it means
to be a prime number. And what you'll
see in this video, or you'll hopefully
see in this video, is it's a pretty
straightforward concept. But as you progress through
your mathematical careers, you'll see that there's actually
fairly sophisticated concepts that can be built on top of
the idea of a prime number. And that includes the
idea of cryptography. And maybe some of the encryption
that your computer uses right now could be
based on prime numbers. If you don't know
what encryption means, you don't have to worry
about it right now. You just need to know the prime
numbers are pretty important. So I'll give you a definition. And the definition might
be a little confusing, but when we see
it with examples, it should hopefully be
pretty straightforward. So a number is prime if
it is a natural number-- and a natural number, once
again, just as an example, these are like the numbers 1, 2,
3, so essentially the counting numbers starting
at 1, or you could say the positive integers. It is a natural number divisible
by exactly two numbers, or two other natural numbers. Actually I shouldn't
say two other, I should say two
natural numbers. So it's not two other
natural numbers-- divisible by exactly
two natural numbers. One of those numbers is itself,
and the other one is one. Those are the two numbers
that it is divisible by. And that's why I didn't
want to say exactly two other natural numbers,
because one of the numbers is itself. And if this doesn't
make sense for you, let's just do some
examples here, and let's figure out if some
numbers are prime or not. So let's start with the smallest
natural number-- the number 1. So you might say, look,
1 is divisible by 1 and it is divisible by itself. You might say, hey,
1 is a prime number. But remember, part
of our definition-- it needs to be divisible by
exactly two natural numbers. 1 is divisible by only one
natural number-- only by 1. So 1, although it might be
a little counter intuitive is not prime. Let's move on to 2. So 2 is divisible by
1 and by 2 and not by any other natural numbers. So it seems to meet
our constraint. It's divisible by exactly
two natural numbers-- itself, that's 2 right there, and 1. So 2 is prime. And I'll circle
the prime numbers. I'll circle them. Well actually, let me do
it in a different color, since I already used
that color for the-- I'll just circle them. I'll circle the
numbers that are prime. And 2 is interesting
because it is the only even number
that is prime. If you think about it,
any other even number is also going to be
divisible by 2, above and beyond 1 and itself. So it won't be prime. We'll think about that
more in future videos. Let's try out 3. Well, 3 is definitely
divisible by 1 and 3. And it's really not divisible
by anything in between. It's not divisible by 2, so
3 is also a prime number. Let's try 4. I'll switch to
another color here. Let's try 4. Well, 4 is definitely
divisible by 1 and 4. But it's also divisible by 2. 2 times 2 is 4. It's also divisible by 2. So it's divisible by three
natural numbers-- 1, 2, and 4. So it does not meet our
constraints for being prime. Let's try out 5. So 5 is definitely
divisible by 1. It's not divisible by 2. It's not divisible by 3. It's not exactly divisible by 4. You could divide them into it,
but you would get a remainder. But it is exactly
divisible by 5, obviously. So once again, it's divisible
by exactly two natural numbers-- 1 and 5. So, once again, 5 is prime. Let's keep going,
just so that we see if there's any
kind of a pattern here. And then maybe I'll
try a really hard one that tends to trip people up. So let's try the number. 6. It is divisible by 1. It is divisible by 2. It is divisible by 3. Not 4 or 5, but it
is divisible by 6. So it has four natural
number factors. I guess you could
say it that way. And so it does not have
exactly two numbers that it is divisible by. It has four, so it is not prime. Let's move on to 7. 7 is divisible by 1, not 2,
not 3, not 4, not 5, not 6. But it's also divisible by 7. So 7 is prime. I think you get the
general idea here. How many natural
numbers-- numbers like 1, 2, 3, 4, 5, the numbers
that you learned when you were two years old, not including 0,
not including negative numbers, not including fractions and
irrational numbers and decimals and all the rest, just regular
counting positive numbers. If you have only two
of them, if you're only divisible by yourself and
one, then you are prime. And the way I think
about it-- if we don't think about the
special case of 1, prime numbers are kind of these
building blocks of numbers. You can't break
them down anymore they're almost like the
atoms-- if you think about what an atom is, or
what people thought atoms were when
they first-- they thought it was kind of the
thing that you couldn't divide anymore. We now know that you
could divide atoms and, actually, if
you do, you might create a nuclear explosion. But it's the same idea
behind prime numbers. In theory-- and in prime
numbers, it's not theory, we know you can't
break them down into products of
smaller natural numbers. Things like 6-- you could
say, hey, 6 is 2 times 3. You can break it down. And notice we can break it down
as a product of prime numbers. We've kind of broken
it down into its parts. 7, you can't break
it down anymore. All you can say is that
7 is equal to 1 times 7, and in that case, you really
haven't broken it down much. You just have the 7 there again. 6 you can actually
break it down. 4 you can actually break
it down as 2 times 2. Now with that out of the way,
let's think about some larger numbers, and think about whether
those larger numbers are prime. So let's try 16. So clearly, any number is
divisible by 1 and itself. Any number, any natural
number you put up here is going to be
divisible by 1 and 16. So you're always
going to start with 2. So if you can find anything
else that goes into this, then you know you're not prime. And 16, you could have 2 times
8, you could have 4 times 4. So it's got a ton
of factors here above and beyond
just the 1 and 16. So 16 is not prime. What about 17? 1 and 17 will
definitely go into 17. 2 doesn't go into 17. 3 doesn't go. 4, 5, 6, 7, 8, 9 10, 11--
none of those numbers, nothing between 1
and 17 goes into 17. So 17 is prime. And now I'll give
you a hard one. This one can trick
a lot of people. What about 51? Is 51 prime? And if you're
interested, maybe you could pause the
video here and try to figure out for yourself
if 51 is a prime number. If you can find anything
other than 1 or 51 that is divisible into 51. It seems like, wow, this is
kind of a strange number. You might be tempted
to think it's prime. But I'm now going to give you
the answer-- it is not prime, because it is also
divisible by 3 and 17. 3 times 17 is 51. So hopefully that
gives you a good idea of what prime numbers
are all about. And hopefully we can
give you some practice on that in future videos or
maybe some of our exercises.