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### Course: Pre-algebra>Unit 1

Lesson 3: Prime factorization

# Prime factorization exercise

Learn to find the prime factorization of any number by breaking it down into its prime factors. Understand the difference between prime and composite numbers. Learn from guided examples of finding the prime factorization of 36, 30, and 73. 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 :( .
• 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
• why is it called prime factorization? why couldn't it be called prime multiples?
• 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.
• How can you use Prime Factorization in the real world?
• 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
• 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?
• 1 is not a prime number because it only has one factor but prime have exactly 2
• 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.
• 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.
• 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?
• 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.
• Do you understand how to do the factor tree??
• I have a question. I don't get it that on the number 36 the "adding the digits" doesn't work. Because 3+6=9 and nine isn't divisible by two, yet 36 IS divisible by 2.
• this only works for numbers that are a multiple of 3
• 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
• 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!