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# Basic Rhythms

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### Learn about Number Bases

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Age 16 to 18

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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?

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

We are used to writing numbers in base ten, using 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Eg. 75 means 7 tens and five units. This article explains how numbers can be written in any number base.