Let $p$ be a prime number, and put

$$a_{k}=k p, \quad N_{k}=a_{k}^{p}-1 \quad(k=1,2, \ldots)$$

Prove that $a_{k}$ has exact order $p$ modulo $N_{k}$ for all $k \geqslant 1$, and deduce that $N_{k}$ must be divisible by a prime $q$ with $q \equiv 1 \quad(\bmod p)$. By making a suitable choice of $k$, prove that there are infinitely many primes $q$ with $q \equiv 1 \quad(\bmod p)$.