No Way Back Home

We are presented a key-exchange protocol where there is some $v$ generated by $p * x$, where $p$ is one of the two primes given, $x$ is a random number generated. We are also given the following: $$ vka = v \times k_A \mod n $$ $$ vkakb = vka \times k_B \mod n $$ $$ vkb = v \times k_B \mod n $$ Obviously we can calculate $k_A$ by doing $(vkb)^{-1} \times vkakb$, but unfortunately $v$ is a multiple of $p$ that is not possible....