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# Prime factorization

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Manager
Joined: 11 Aug 2012
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Schools: HBS '16, Stanford '16
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20 Oct 2012, 12:11
I have a conceptual question:

Does each positive integer have a unique combination of prime factors when we make its prime factorization?
In other words, when we calculate the prime factorization of a number, will we get always the same answer (combination)?, or there could be different sets of prime numbers (including repetitions) whose product is the same number?

For example, 100 = $$2^2$$*$$5^2$$ , in this case is the only set of prime factors whose product is 100. However, I wonder whether in other numbers the opposite is possible.

Thanks!
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20 Oct 2012, 12:25
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To put it very simply. The answer is yes... There is only one unique combination of prime factors for each number. Obviously, no prime number is a multiple or a divisor of another prime number. So there is no other possible combination.
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20 Oct 2012, 12:41
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danzig wrote:
I have a conceptual question:

Does each positive integer have a unique combination of prime factors when we make its prime factorization?
In other words, when we calculate the prime factorization of a number, will we get always the same answer (combination)?, or there could be different sets of prime numbers (including repetitions) whose product is the same number?

For example, 100 = $$2^2$$*$$5^2$$ , in this case is the only set of prime factors whose product is 100. However, I wonder whether in other numbers the opposite is possible.

Thanks!

Hi danzig,

There has to be a unique solution only. If you come across more than one combination of factors for a composite number, for sure there is a composite number in the factors and it is not a prime numbers' group. As answered by MacFauz, it is the property of prime numbers that they do not have other roots and cannot be further factorized, thus there can be only one combination...
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20 Oct 2012, 13:00
Thank you buddies!
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25 Oct 2012, 21:52
Expert's post
danzig wrote:
I have a conceptual question:

Does each positive integer have a unique combination of prime factors when we make its prime factorization?
In other words, when we calculate the prime factorization of a number, will we get always the same answer (combination)?, or there could be different sets of prime numbers (including repetitions) whose product is the same number?

For example, 100 = $$2^2$$*$$5^2$$ , in this case is the only set of prime factors whose product is 100. However, I wonder whether in other numbers the opposite is possible.

Thanks!

A lot of further concepts depend on a thorough understanding on this.

Think of prime factors as basic indivisible building blocks for numbers.
2 - Red block
3 - Yellow block
5 - Green block
and so on..

To make 4, you need 2 red blocks.
To make 100, you need 2 red and 2 green blocks
To make 300, you need 2 red, 2 green and 1 yellow block.
Conversely, whenever you take 1 red and 1 yellow block, you will get 6.
Whenever you take 1 yellow block and 1 green block, you will get 15.
and so on...

Whenever you take 2 red and 2 green blocks, you will always get a 100. Whenever you try to break down 100, you will always get 2 red and 2 green blocks.

You can write 100 as 10*10 but each 10 is made up of 1 red and 1 green block so finally you have 2 red and 2 green blocks only.
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Re: Prime factorization   [#permalink] 25 Oct 2012, 21:52
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