How to find the radix (base) of a number given its representation in another radix (base)?

Question

What's the method to find the base of any given number?

E.g. find $r$ such that $(121)_r=(144)_8$, where $r$ and $8$ are the bases.

So how do I find the value of $r$?

Answer 1

By definition $$144_8=1\cdot8^2+4\cdot8^1+4\cdot8^0=64+32+4=100\;,$$ and $$121_r=1\cdot r^2+2\cdot r^1+1\cdot r^0=r^2+2r+1\;.$$ To find $r$, just solve the quadratic equation $$r^2+2r+1=100$$ by whatever method you find most convenient; the slickest method is probably to notice that $r^2+2r+1=(r+1)^2$, so $(r+1)^2=100=10^2$.