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## Class 10 math (India)

### Course: Class 10 math (India) > Unit 1

Lesson 2: Fundamental theorem of arithmetic# Prime factorization exercise

Created by Sal Khan.

## Want to join the conversation?

- What if it were a very large number say 757, that means that you'd have to see if it's evenly divisible by every prime number from 2 to, well half of 757? Seems like an awful lot of work :( .(85 votes)
- You only have to go up to the square root of 757. Roughly estimating, 25 squared is 625 and 30 squared is 900, so it will be in between, but since 29 is the only prime and that is too close to 30 (29 squared will be 900 - 30 - 29 = 841), we actually only need to go up to 23. So the numbers we need to check are 2, 3, 5, 7, 11, 13, 17, 19 and 23.

We can tell it won't be divisible by 2 since it's not even; it won't be divisible by 3 since the digits don't sum to a multiple of 3 (https://www.khanacademy.org/math/arithmetic/factors-multiples/divisibility_tests/v/the-why-of-the-3-divisibility-rule); and we know it's not divisible by 5 as it doesn't end in a 5. So that leaves us with just**six**numbers to check, and those we would have to check one by one, which I agree is a pain.

In fact there are also tricks to test whether something is divisible by 7 (http://en.wikipedia.org/wiki/Divisibility_rule#Divisibility_by_7) and 11 (http://www.wikihow.com/Check-Divisibility-of-11), but they require a bit more work. For 7, I would just think, if 757 were divisible by 7, then if we subtract multiples of 7 the results would have to be divisible by 7. Subtracting 7, gets us 750, subtracting 700, gets us 50 and that's not divisible by 7, so 757 is also not. So now we're down to**four**numbers(110 votes)

- why is it called prime factorization? why couldn't it be called prime multiples?(19 votes)
- A factor is a number that divides your number. So 4 is a factor of 8 for example. A prime factors is a prime numbers that divides your number. Hence, 4 is not a prime factor of 8, but 2 is. Prime factorization now is the process of splitting a number into its prime factors.(10 votes)

- How can you use Prime Factorization in the real world?(8 votes)
- This is a really good question!

Prime factorisation is essentially the act of breaking large numbers into their constituent building blocks. Natural numbers are made up of these prime factors and so to really understand them we need to be able to take them to pieces (with prime factorisation). You could think of it like taking a car engine to pieces and putting them together again, to understand how the engine works.

One important use of prime factorisation is in making (or breaking!) encrypted data. Encryption of data keeps it secure and stops people other than the intended recipient from looking at the data. We all rely on data encryption, especially people that handle sensitive data such as governments and businesses. If data encryption stopped working then it would make our societies very difficult to run!

Other applications of prime numbers relate to mathematical theory. You can read more about them here https://en.wikipedia.org/wiki/Prime_number#Public-key_cryptography(7 votes)

- For the exercise - prime factorization - say you have a number such as 5 and you put in the answer box 1*5 it says it is wrong. I am not sure why that is because 1x5=5. So, why does it count 1*5 as wrong?(4 votes)
- 1 is not a prime number because it only has one factor but prime have exactly 2(4 votes)

- Might sound like a stupid question but what is the prime factorisation of 1??

I mean, 1=1 but 1 is not a prime number.(2 votes)- You are correct. 1 is not a prime number. It is also not a composite number. It has only one factor, itself. This makes it unique. Because it is not a composite number, it does not have a prime factorization.(5 votes)

- Do we always have to go from the smallest prime factor to the largest prime factor when prime factorizing? I had to prime factorize 3,628,800 and I could solve it when I started from the greatest prime factor, which was 7. Am I supposed to do the prime factorizing from the smallest number even when prime factorizing big numbers such as 3,628,800?(2 votes)
- It is not necessary to start prime factorizing from the smallest number.

You might do that in any way, since it does not change the answer(each number has a unique prime factorization).(1 vote)

- Hi! How y'all? Please help me with the factor tree, and also finding it. (The GCF). It's so hard for me.(2 votes)
- Do you understand how to do the factor tree??(3 votes)

- Is there any easier way? This would be VERY time consuming if it were something like 142, 386.(2 votes)
- How bout what is the prime factorization of 3240.... its to big and i havent memorized that chart thing that tells you which numbers can be divisible if that even makes sense(2 votes)
- My answer's a little long, but it's a bunch of pretty easy steps:

First, notice that 3240 ends in a zero, which makes it easier, because you can divide anything that ends in a zero by 10: so now the number becomes: 10 X 324.

Suddenly that big number isn't so big any more.

Next, do the prime factorization of 10: 5 X 2 X 324.

So all you have to deal with now is the 324. Anything that ends in an even number can be divided by 2:

5 X 2 X 2 X 162.

The 162 ends in an even number, so you can divide again by 2:

5 X 2 X 2 X 2 X 81.

If you know your multiplication tables, you recognize 81 as the product of 9 X 9:

5 X 2 X 2 X 2 X 9 X 9.

And finally, you do the prime factorization of those 9s, which is 3 X 3, for each 9:

5 X 2 X 2 X 2 X 3 X 3 X 3 X 3 = 3240.

So anytime you're facing a big number, look for easy ways to make it smaller -- Is it even? Then you can divide by 2 -- Does it end in 5? Then you can divide by 5 -- Does it end in 0? Then you can divide by 10. If none of those numbers help, you can try dividing by other small numbers. The key is making the big numbers smaller!(2 votes)

- What if the number is a three digit number and it's an even number?(2 votes)
- You would just keep on going until you've found the answer.(1 vote)

## Video transcript

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.