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# Basic Rhythms

##### Age 16 to 18Challenge Level

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?