TSTST 2014 Problem 6

Suppose we have distinct positive integers $a, b, c, d$ and an odd prime $p$ not dividing any of them, and an integer $M$ such that if one considers the infinite sequence $$\begin{gathered} c a-d b \\ c a^{2}-d b^{2} \\ c a^{3}-d b^{3} \\ c a^{4}-d b^{4} \end{gathered}$$ and looks at the highest power of $p$ that divides each of them, these powers are not all zero, and are all at most $M$. Prove that there exists some $T$ (which may depend on $a, b, c, d, p, M)$ such that whenever $p$ divides an element of this sequence, the maximum power of $p$ that divides that element is exactly $p^{T}$.