Question 6
For each $1 \leqslant i \leqslant 9$ and $T \in \mathbb{N}$, define $d_{i}(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$.
Show that there are infinitely many $T \in \mathbb{N}$ such that there are precisely two distinct values among $d_{1}(T), d_{2}(T), \ldots, d_{9}(T)$.
Show that there are infinitely many $T \in \mathbb{N}$ such that there are precisely two distinct values among $d_{1}(T), d_{2}(T), \ldots, d_{9}(T)$.