My naive question may actually lead to something interesting.
Let $\Phi_m(x)$, $\Phi_n(x)$ be cyclotomic polynomials, $m<n$. These polynomials are relatively prime and so there are polynomials $s(x), t(x)$ with rational coefficients such that $s\Phi_m+t\Phi_n=1$. Let $d$ be the least common denominator of the coefficients of the polynomials $s,t$ (of minimal degrees, $s,t$ obtained by the Euclidean algorithm). A Maple experiment shows that the following conjectures may be true.
Conjecture 1. If the number $d$ is not equal to $1$ then $m$ divides $n$ and $n/m$ is a prime power.
Conjecture 2. If $n=mp^k$ where $p$ is prime, then $d=p$.
Question. Are these conjectures true?
Motivation: $\Phi_m$ and $\Phi_n$ are co-prime in the ring of integers modulo any number that is co-prime with $d$. In particular, if $d=1$, then the reductions of $\Phi_m$ and $\Phi_n$ modulo any number would be relatively prime.