Proof: no fractions that can't be written in lowest term with Well Ordering Principle

Question

My question is the exact same question as the one in this post but I commented on it but it's from a year ago so I just wanted to bump it and see if I could get a response:

Prove that there's no fractions that can't be written in lowest term with Well Ordering Principle

I don't understand what the text means by "so by definition of C, there is an integer $n_0 > 0$ such that the fraction ${m_0\over n_0}$ cannot be written in lowest terms. This means that $m_0$ and $n_0$ must have a common prime factor, $p >1$."

If ${m_0\over n_0}$ cannot be written in lowest terms why is it true that $m_0$ and $n_0$ must have a common prime factor (which excludes 1 since 1 isn't prime)?

Thanks.

Answer 1

That statement is confusing, since the conclusion doesn't need the full strength of the assumptions. Here's what they really mean.

1. If $\frac{m_0}{n_0}$ cannot be written in lowest terms, then certainly $\frac{m_0}{n_0}$ is not already written in lowest terms.
2. And if $\frac{m_0}{n_0}$ is not written in lowest terms, then $m_0$ and $n_0$ have a common prime factor.

Answer 2

Def. From Wolfram: http://mathworld.wolfram.com/ReducedFraction.html

As Adam Hughes said, they're basically doing a proof by contradiction since lowest terms means that the $GCD(m_0, n_0)=1 $ and both numerator and denominator have no common factors except for 1. So assuming otherwise means that they share a common factor other than 1.

The Well Ordering Principle hints that if there was a least common factor other than 1 then the factor would have a prime component of 2 (smallest prime) or greater in it's prime decomposition and the fraction wouldn't be written in lowest terms.

Their argument could probably be stated better as Chris said.