Putnam 1994 B4

For $n \geq 1$, let $d_n$ be the greatest common divisor of the entries of $A^n - I$, where $$A = \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix} \text{ and } I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.$$ Show that $\lim_{n \to \infty} d_n = \infty$.