TITLE: Cyclotomic polynomials 2
QUESTION [16 upvotes]: 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.

REPLY [10 votes]: This note (see theorem 2.2) attributes this result to

Diederichsen, Fritz-Erdmann,
  "Über die Ausreduktion ganzzahliger Gruppendarstellungen bei arithmetischer Äquivalenz", Abh. Math. Sem. Hansischen Univ. 13, (1940). 357–412

but also points out to pp. 550-554 in Curtis/Reiner "Methods of representation theory, with applications to finite groups and orders" vol.1.