For each integer $k \ge 2$, the decimal expansions of the numbers $1024, 1024^2, \dots, 1024^k$ are concatenated, in that order, to obtain a number $X_k$. For example, $X_2 = 10241048576$. If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n \ge 2$ is an integer.

Let's think step by step and output the final answer within \boxed{}.
user