Prime Factorization (Arithmetic)

In summary, the conversation discusses the assumption that 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 and none of the p's are equal to any of the q's. The participants question the validity of this assumption based on the Fundamental Theorem of Arithmetic, which states that each number n \geq 2 has a unique prime factorization. They suggest that the p's and q's must be equal if their product is the same, and explore possible exceptions to this rule.
  • #1
cheiney
11
0

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[itex]\geq[/itex]2 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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  • #2
Is this literally the statement of your homework question?
 
  • #3
I'm with you. The question doesn't make any sense to me either.
 
  • #4
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.
 
  • #5
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.
 

FAQ: Prime Factorization (Arithmetic)

1. What is Prime Factorization?

Prime factorization is the process of breaking down a number into its prime factors, which are numbers that can only be divided by 1 and itself with no remainder. This is an important concept in arithmetic and is used to simplify fractions and find the greatest common factor of two numbers.

2. How do I find the prime factors of a number?

To find the prime factors of a number, you can use a factor tree or divide the number by its smallest prime factor and continue dividing the resulting factors until all factors are prime. For example, to find the prime factors of 24, you can divide by 2 to get 12, then divide 12 by 2 to get 6, and finally divide 6 by 2 to get 3. The prime factors of 24 are 2, 2, 2, and 3.

3. Why is prime factorization important?

Prime factorization is important because it allows us to simplify fractions, find the greatest common factor of two numbers, and find the prime factorization of large numbers. It is also used in cryptography, a field that deals with data encryption and security.

4. Can all numbers be prime factorized?

Yes, all numbers can be prime factorized. Even prime numbers, which are only divisible by 1 and itself, can be considered as having a prime factorization of itself multiplied by 1. For example, the prime factorization of 17 is 17 x 1.

5. What is the difference between prime factorization and prime numbers?

Prime factorization is the process of breaking down a number into its prime factors, while prime numbers are numbers that can only be divided by 1 and itself with no remainder. In other words, prime factorization is a method used to find the prime numbers that make up a given number.

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