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Let $\mathbb{N} = \{0, 1, 2, \cdots\}$.

Fix an integer $b \ge 2$. For $N \in \mathbb{N}$ we say that $N$ is a base-$b$ dead end if $N$ is square-free and for every digit $d \in \{0, 1, \cdots, b-1\}$, the integer $bN+d$ is not square-free.

Now define the ``asymptotic density'' for base $b$:
\[D_b := \lim_{X \to \infty} \frac{\#\{N \le X : N \text{ is a base-b dead end}\}}{X}\]

Find an explicit form for $D_b$, and then prove it.

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