Prime Factorization (Arithmetic)

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Homework Help Overview

The discussion revolves around the concept of prime factorization as stated in the Fundamental Theorem of Arithmetic. The original poster presents a scenario where a number n can be expressed as a product of two distinct sets of prime factors, questioning the validity of the assumption that the primes in each set are not equal.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of the Fundamental Theorem of Arithmetic, questioning the assumption that the primes in the two factorizations can be different. Some suggest considering the conditions under which the statement might hold, while others express confusion about the question's validity.

Discussion Status

The discussion is ongoing, with participants sharing their interpretations and questioning the assumptions made in the problem statement. Some have offered alternative perspectives on the nature of prime numbers and factorization, but no consensus has been reached.

Contextual Notes

There is a noted ambiguity regarding the conditions under which n is defined, particularly whether n is constrained to be greater than or equal to 2, which may affect the validity of the assumptions being discussed.

cheiney
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Homework Statement



Assume n = p_1*p_2*p_3*...*p_r = q_1*q_2*q_3*...*q_s, where the p's and q's are primes. We can assume that none of the p's are equal to any of the q's. Why?

Homework Equations





The Attempt at a Solution



I am completely stuck on this. My understanding of the Fundamental Theorem of Arithmetic is that each number n\geq2 has a unique prime factorization. So how could we possibly assume that the p's aren't equal to the q's?
 
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Is this literally the statement of your homework question?
 
I'm with you. The question doesn't make any sense to me either.
 
Office_Shredder said:
Is this literally the statement of your homework question?

Yes. The only 2 approaches I could really think of is that, since they didn't specify that n is greater than or equal to 2, so if all of q_j are negative and if j is an even number, then it would hold up. The other approach would be to describe n as a member of a set with "primes" in the sense that they cannot be divisible by other numbers in the set other than 1 and itself.
 
I would be inclined to say that "prime" implies positive and so the "unique factorization property" says, to the contrary of what this purports, that if p_1*p_2*p_3*...*p_r = q_1*q_2*q_3*...*q_s, then the "p"s and "q"s must be equal.
 

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