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Basic Rhythms

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Notice the following pattern, written in base 10:

$$ \begin{eqnarray} 987654321&=&8\times 123456789 &+&9\\ 98765432&=&8\times 12345678 &+&8 \\ 9876543&=&8\times 1234567 &+&7 \\ 987654&=&8\times 123456 &+&6\\ &\vdots & \\ 9&=&8\times 1&+&1 \end{eqnarray} $$

This patterns also holds in bases other than $10$. For example, in base $4$ we have $321 = 2 \times 123 + 3 $, and so on.

Why is this the case?