This is an exercise from a math chapter called number systems where I learn how to convert decimal numbers to other number bases like binary and hexadecimal.
Here I need to discover which base number system is $23^2 = 562$. I know that the resulting base number system is $7$. Since $23$ septenary = $17$ decimal, and $17^2$ decimal = $289$, I can confirm that $289$ decimal = $562$ septenary.
The problem is that I must discover $7$ using equations and not with trial-and-error. Thank you!
Best Answer
So that this question can be marked as answered:
In base $b$, $23^2 = (2b + 3)^2 = 4b^2 + 12b + 9$. Equate this to $5b^2 + 6b + 2$ to get $b^2 - 6b - 7 = 0$, which can now be solved to get $b = 7$.