  Let $\Bbb F_q$ be a finite field with $q$ elements. Let $n$ be a positive
integer with radical $rad(n)$, namely, the product of distinct prime divisors
of $n$. If the order of $q$ modulo $rad(n)$ is either 1 or a prime, then the
irreducible factorization and a counting formula of irreducible factors of
$x^n-1$ over $\Bbb F_q$ were obtained by Mart\'{\i}nez, Vergara, and Oliveira
(Des Codes Cryptogr 77 (1) : 277-286, 2015) and Wu, Yue, and Fan (Finite Fields
Appl 54: 197-215, 2018). In this paper, we explicitly factorize $x^{n}-1$ into
irreducible factors in $\Bbb F_q[x]$ and calculate the number of the
irreducible factors when the order of $q$ modulo $rad(n)$ is a product of two
primes.
