# Prime Factorization (Arithmetic)

• cheiney

## 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?

## The Attempt at a Solution

I am completely stuck on this. My understanding of the Fundamental Theorem of Arithmetic is that each number n$\geq$2 has a unique prime factorization. So how could we possibly assume that the p's aren't equal to the q's?

Is this literally the statement of your homework question?

I'm with you. The question doesn't make any sense to me either.

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.