Contrapositive and Contradiction · Hard problem

Primes of the form $4k + 3$: suppose there were only finitely many, $p_1, \ldots, p_k$, and let $N = 4p_1\cdots p_k - 1$. For the list $3, 7, 11$ this gives $N = 923$. Which prime factor of $N$ has the form $4k + 3$?