 Solution

Suppose only finitely many primes are congruent to five, and list them as $p_1,\dots ,p_k$. Consider $N=6p_1\cdots p_k-1>1$. It is divisible by neither two nor three, so every prime divisor is congruent to one or five modulo six. If all its prime factors were congruent to one, their product would be congruent to one by part (i), whereas $N\equiv 5$. Hence some prime divisor $q$ has $q\equiv 5(\mathrm{mod}6)$.

But no listed $p_i$ divides $N$, since $N\equiv -1(\mathrm{mod}{p}_i)$. This gives a new such prime, a contradiction. Therefore infinitely many primes are congruent to five modulo six. The argument does not require the specially constructed integer itself to be prime.