Elementary Number Theory Chapter 3.3 Page 54, Why "But $r_{i}$ cannot be found among the listing $q_{1}, q_{2}, ...$"?(David M. Burton's 7th Edition)
Background
This question arises from David M. Burton's proof for "there are an infinite number of primes of the form $4n + 3$".
Explanation
Think about this, if $r_{i}$ is one of $q_{i}$, then $r_{i} \mid 4q_{1}q_{2} \cdots q_{s}$, and we know $r_{i} \mid N$. Then by Theorem 2.2(g), $r_{i} \mid 4q_{1}q_{2} \cdots q_{s} - N = 1$. This leads to contradiction as $r_{i}$ is $N$'s prime factorization.