I have a calculator that I found recently that has a base 5 option. It has options labelled HEX, BIN, DEC, OCT, and PEN. My son set it on PEN, and no numbers above 4 worked anymore. Playing around with it, it was clearly base 5. (E.g. 22 divided by 3 is 4).
Why does any calculator have base 5? The other bases are commonly used in computer science, but I can't think of the slightest reason why anyone would need base 5 on a calculator.
The calculator is by Sharp, with the number EL-W535 and the word Writeview.
Best Answer
Apparently, some computers in the 50's used a system to describe the decimal system which had base five digits, and another bit to represent if it was between $0-4$ or between $5-9$, known as a biquinary system.