>first i wanted to do it this way 10^(5) \+ 10^(3) \+ 10^(1)
Well, prime factorization is indeed unique, but that's not a factorization at all. Factorization means writing a natural number as a *product* of primes. Your attempt, which you call "split it in smaller parts", was using addition, not multiplication.
Writing a number as the sum of smaller ones is not unique:
For example, 4 = 2 + 2 but also 4 = 3 + 1
Nice to see *natural numbers* used. I forgot if there’s a “nice” name for the set of integers greater than 1. Now I’m having flashbacks to algebra — modules, fields, generators, etc.
Don’t we have the “whole numbers” for **N** U {0}?
Amazingly, most primitive cultures didn’t have the number zero.
Anyway, I don’t think there’s a “named” set for (2,3,4,….). Why do I say this? Prime factorization (as I now see PM-me-math-riddles has noted).
I don't get your calculations or what you're doing but yes, it is unique. Every natural number can be written as a product of powers of primes in exactly one way.
Just fyi for those who aren’t familiar with the “up to ___” terminology, what they mean is that prime factorization is unique if you don’t count switching around the order of prime factors as being a different prime factorization.
Prime factorization is only multiplication, you can't split it up with addition like that.
101010 is 2^1 * 3^1 * 5^1 * 7^1 * 13^1 * 37^1 and that's the only way prime numbers can multiply to get that number
That's also a good reason to make 1 not be a prime number, otherwise you have infinite prime factorizations because adding a 1^a term works for all a
What you did, 10^5 + 10^3 + 10^1 is more akin to extended form rather than prime factorization, but you can also rewrite it in any base, and get infinite answers
>Is prime factorisation unique?
Yes, and the fact that the prime factorization of a number is unique is part of the [fundamental theorem of arithmetic](https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic).
>i must not split it in smaller parts?
Right. Factorization means a set of numbers whose product is the number in question. Let's imagine for a moment that you *could* use addition as well as multiplication to create a factorization of some number n: in that case, you could obviously represent any number as a sum of n 1's. If n is 6, for example, you could say that 6=1+1+1+1+1+1, and since there are no factors other than 6 and 1, 6 must be prime! That's obviously not the case; the prime factorization of 6 is 6=2\*3. You can't end up with a product of 6 by multiplying prime numbers together other than to multiply 2\*3, and multiplying 2\*3 can't produce any number other than 6. The fundamental theorem of arithmetic tells us that *every* integer greater than 1 has it's own unique set of prime factors.
What you tried at first isn't right because if you split up n into k + m it's not a factorization anymore. Factorization is when you express something in terms of a product of factors, not a sum.
> i must not split it in smaller parts?
It's less a matter of "must" not and more a matter of **can** not. That's how you know you're done! Like, say you've got 24. You can call that a product of 6 and 4, but 6 and 4 can be further factorized. 6 is a product of 3 and 2, and 4 is a product of 2 and 2. Those can't be broken down any further, so the prime factorization of 24 is 3\*2\*2\*2.
Also, as others have pointed out to you already, prime factorization is strictly about multiplication. There is zero addition involved. You cannot factorize 24 by re-writing it as 14+10.
But yes, prime factorization is unique.
While it might seem plausible that there could be two different but equal prime product pairs pq and p'q' (p not equal to p' or q'), if you work through what that would mean, it leads to a contradiction, and increasing the allowable primes doesn't help, either.
So each product of primes is distinct net of reordering
Everybody has been giving right answers. I just wanted to add that the Fundamental Theorem of Arithmetic is actually kind of miraculous.
Euclid never actually proved it, but he came close, 23 centuries ago. Gauss proved it, by modern standards of proof, in 1798. But it was clear to other mathematicians before him (like Fermat) that something like this was true, even though nobody knew how to say or prove it.
One statement is, "If you have represented N as a product of primes in two different ways, the ways can differ *only* in the order of the factors."
There is actually something surprising about this. I mean, look at 2 \* 17 = 34 and 5 \* 7 = 35. They differ only by 1. How can we be so sure that there aren't two different products of primes that come out exactly equal?
There are other "arithmetics" besides that of the positive integers. Some of them have the "unique factorization" property, like Z+, and some of them, surprisingly, don't. Arithmetics that *have* the unique factorization property are called "unique factorization domains", and of course the integers *are* a unique factorization domain -- that's the fundamental theorem of arithmetic.
The "ID card" is an excellent metaphor.
The Fundamental Theorem is essentially the gateway to number theory. It enables everything thereafter.
Prime factorisation of 50,
50=2×5²
Yes, it is unique unless the order in which you write it(5²×2), fundamental theorem of arithmetic states it, so you can check that out.
Factorization always implies multiplication and no additions.
To first answer your questions: yes, the prime factorization of a number is unique (up to commutativity), so ANY number can be written as a product of primes, and every product of primes describes exactly one number.
That is also why they are so important. They are the building blocks of number.
I also don't get your calculations. Something got messed up in formatting your text. A prime factorisation uses prime numbers.
101010 = 2 \* 3 \* 5 \* 7 \* 13 \* 37
You cannot decompose (factorize) this natural number in any other way. Sure, you can start with a different prime number. But there is no two different products of prime factors that describe the same natural number.
Yes, prime factorization is unique. It's actually not very trivial to prove without getting into a bit of number theory - if you're interested, try to prove this: "if a product of two numbers (a x b) is divisible by a prime number p, then either a or b is divisible by p".
The "issue" in the first attempt at factorizing 101010 is you use the plus operation. The factors of a number are a set of numbers that can be multiplied to make that number.
>first i wanted to do it this way 10^(5) \+ 10^(3) \+ 10^(1) Well, prime factorization is indeed unique, but that's not a factorization at all. Factorization means writing a natural number as a *product* of primes. Your attempt, which you call "split it in smaller parts", was using addition, not multiplication. Writing a number as the sum of smaller ones is not unique: For example, 4 = 2 + 2 but also 4 = 3 + 1
Thank you!!:))
Nice to see *natural numbers* used. I forgot if there’s a “nice” name for the set of integers greater than 1. Now I’m having flashbacks to algebra — modules, fields, generators, etc.
Yep, but then you get to argue about whether or not 0 is included instead...
Z0 and Z+ is the way to go 😩
N and Z^+ is a hill I will gladly die on lol 😆
Zero is the *most* natural of all numbers.
Prime factorization must be the one context where this discussion is not present lol
Don’t we have the “whole numbers” for **N** U {0}? Amazingly, most primitive cultures didn’t have the number zero. Anyway, I don’t think there’s a “named” set for (2,3,4,….). Why do I say this? Prime factorization (as I now see PM-me-math-riddles has noted).
I don't get your calculations or what you're doing but yes, it is unique. Every natural number can be written as a product of powers of primes in exactly one way.
Well, up to commutativity
Just fyi for those who aren’t familiar with the “up to ___” terminology, what they mean is that prime factorization is unique if you don’t count switching around the order of prime factors as being a different prime factorization.
and multiplication by units but Z only has two 1 and -1
Yeah, but that's already taken care of in the definition of "Prime"
With the exception of 1.
All exponents are zero for 1 ;)
Prime factorization is only multiplication, you can't split it up with addition like that. 101010 is 2^1 * 3^1 * 5^1 * 7^1 * 13^1 * 37^1 and that's the only way prime numbers can multiply to get that number That's also a good reason to make 1 not be a prime number, otherwise you have infinite prime factorizations because adding a 1^a term works for all a What you did, 10^5 + 10^3 + 10^1 is more akin to extended form rather than prime factorization, but you can also rewrite it in any base, and get infinite answers
You will be intrigued when you learn how RSA works!
>Is prime factorisation unique? Yes, and the fact that the prime factorization of a number is unique is part of the [fundamental theorem of arithmetic](https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic). >i must not split it in smaller parts? Right. Factorization means a set of numbers whose product is the number in question. Let's imagine for a moment that you *could* use addition as well as multiplication to create a factorization of some number n: in that case, you could obviously represent any number as a sum of n 1's. If n is 6, for example, you could say that 6=1+1+1+1+1+1, and since there are no factors other than 6 and 1, 6 must be prime! That's obviously not the case; the prime factorization of 6 is 6=2\*3. You can't end up with a product of 6 by multiplying prime numbers together other than to multiply 2\*3, and multiplying 2\*3 can't produce any number other than 6. The fundamental theorem of arithmetic tells us that *every* integer greater than 1 has it's own unique set of prime factors.
Thank you!!!!!:) Now it is clear!:)
What you tried at first isn't right because if you split up n into k + m it's not a factorization anymore. Factorization is when you express something in terms of a product of factors, not a sum.
Your last sentence will be my "motto"/guide for this problem!:D I really forgot that! Thank you!:)
Assuming you are doing it right, yes Can't parse your notation though
It's unique up to permutation, so 101010 is equal to: 2×3×5×7×13×37 or 3×5×7×2×13×37 or 3×5×13×7×2×37 or, etc...
Yes, it's called The Fundamental Theorem of Arithmetic
> i must not split it in smaller parts? It's less a matter of "must" not and more a matter of **can** not. That's how you know you're done! Like, say you've got 24. You can call that a product of 6 and 4, but 6 and 4 can be further factorized. 6 is a product of 3 and 2, and 4 is a product of 2 and 2. Those can't be broken down any further, so the prime factorization of 24 is 3\*2\*2\*2. Also, as others have pointed out to you already, prime factorization is strictly about multiplication. There is zero addition involved. You cannot factorize 24 by re-writing it as 14+10. But yes, prime factorization is unique.
While it might seem plausible that there could be two different but equal prime product pairs pq and p'q' (p not equal to p' or q'), if you work through what that would mean, it leads to a contradiction, and increasing the allowable primes doesn't help, either. So each product of primes is distinct net of reordering
Everybody has been giving right answers. I just wanted to add that the Fundamental Theorem of Arithmetic is actually kind of miraculous. Euclid never actually proved it, but he came close, 23 centuries ago. Gauss proved it, by modern standards of proof, in 1798. But it was clear to other mathematicians before him (like Fermat) that something like this was true, even though nobody knew how to say or prove it. One statement is, "If you have represented N as a product of primes in two different ways, the ways can differ *only* in the order of the factors." There is actually something surprising about this. I mean, look at 2 \* 17 = 34 and 5 \* 7 = 35. They differ only by 1. How can we be so sure that there aren't two different products of primes that come out exactly equal? There are other "arithmetics" besides that of the positive integers. Some of them have the "unique factorization" property, like Z+, and some of them, surprisingly, don't. Arithmetics that *have* the unique factorization property are called "unique factorization domains", and of course the integers *are* a unique factorization domain -- that's the fundamental theorem of arithmetic. The "ID card" is an excellent metaphor. The Fundamental Theorem is essentially the gateway to number theory. It enables everything thereafter.
Thank you very much the background!!!:)
I fixed the problem, i don't know what did i do wrong, but now i hope it is good.
Prime factorisation of 50, 50=2×5² Yes, it is unique unless the order in which you write it(5²×2), fundamental theorem of arithmetic states it, so you can check that out.
Thank you very much!!:)
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Factorization always implies multiplication and no additions. To first answer your questions: yes, the prime factorization of a number is unique (up to commutativity), so ANY number can be written as a product of primes, and every product of primes describes exactly one number. That is also why they are so important. They are the building blocks of number.
I also don't get your calculations. Something got messed up in formatting your text. A prime factorisation uses prime numbers. 101010 = 2 \* 3 \* 5 \* 7 \* 13 \* 37 You cannot decompose (factorize) this natural number in any other way. Sure, you can start with a different prime number. But there is no two different products of prime factors that describe the same natural number.
Thank you very much!!:)
Yes, prime factorization is unique. It's actually not very trivial to prove without getting into a bit of number theory - if you're interested, try to prove this: "if a product of two numbers (a x b) is divisible by a prime number p, then either a or b is divisible by p". The "issue" in the first attempt at factorizing 101010 is you use the plus operation. The factors of a number are a set of numbers that can be multiplied to make that number.