Question 79
For each positive integer $n$ and each integer $i$ ($0 \leq i \leq n$), let $C_n^i \equiv c(n,i) \pmod{2}$, where $c(n,i) \in \{0, 1\}$, and define
$$
f(n,q) = \sum_{i=0}^{n} c(n,i)q^i.
$$
Let $m$, $n$ and $q$ be positive integers with $q+1$ not a power of $2$. Suppose that $f(m,q) \mid f(n,q)$. Prove that
$f(m,r) \mid f(n,r)$ for every positive integer $r$.
$$
f(n,q) = \sum_{i=0}^{n} c(n,i)q^i.
$$
Let $m$, $n$ and $q$ be positive integers with $q+1$ not a power of $2$. Suppose that $f(m,q) \mid f(n,q)$. Prove that
$f(m,r) \mid f(n,r)$ for every positive integer $r$.