Modular equation system I have $N=p\cdot q$ and the following system where I know $A,B,C,D, k$:
$$A = B \cdot q^k \pmod N$$
and
$$C = D \cdot p^k \pmod N$$
Is there an easy way to recover $p$ and $q$?
Is there an easy way to recover $p$ and $q$?

Yes (in all cases except for $B, D$ are both multiples of $N$, or $k=0$, and assuming that $p, q$ are distinct primes)
Let us assume that $B$ is not a multiple of $N=pq$; then:

If $B$ is not a multiple of $p$, then $\gcd(A, N) = q$ (because $A=Bq^k$ is a multiple of $q$ but not $p$); simple division also recovers $p$

If $B$ is a multiple of $p$, then it is not a multiple of $q$ (by assumption); in that case $\gcd(B, N) = p$