Many (but not all) number systems used auxiliary number bases. All number systems that use a main base and an auxiliary base are structured in the same way.
The arithmetic rule for auxiliary number bases is:
A number system with base m can use the number a as auxiliary base if a multiplier b exists such that a b = m |
This rule was applied in many parts of the world. Some examples are:
| number system | main base m | auxiliary base a | multiplier b |
| Sumerian | 60 | 10 | 6 |
| Maya | 20 | 5 | 4 |
| Roman | 10 | 5 | 2 |
Other number systems did not use an auxiliary base. Examples are the Egyptian (base 10) and the Aztec (base 20) number systems.