Base what?

Number Theory Level 4

Suzie McTube decided to invent a unique number system. It's not base 2. It's not base 3, or even base 4, 5, 6 or 7... But it's a combination of all the bases!

It works like this: The first digit (farthest on the right, or "unit's digit") is base 2. The second digit (again from the right) is base 3. The third is base 4 etc. So, for example, the first few numbers are:

\[1, 10, 11, 20, 21, 100, 101, 110, \ldots \]

So, if 21 is the \(5^\text{th}\) number (as shown above), what is the \(99^\text{th}\) number? (99 is in base 10 or decimal)

Inspired by: Ujjwal Rane.

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