<s>[INST] In a numerical system of base $r$, the polynomial $x^2 − 11x + 22 = 0$ has the solutions $3$ and $6$. What is the base r of the system? From Algebra, the statement is equivalent to say that $(x^2− 11x +
22)_{r}$ = $(x − 3)_{r} \cdot (x − 6)_{r}$. Doing operations we arrive at $3 + 6 = 11_{r} = r + 1$, and $(3)(6) = 22_{r} = 2 \cdot 11_{r}$. In any case, $r = 8$.
This is the solution to the problem, but how do I yield to the conclusion that $3 + 6 = 11_{r} = r + 1$? 
 [/INST] If the roots are $3$ and $6$, the equation is $(x-3)(x-6)=x^2-(3+6)x+3\cdot 6=0$  So $11_r=3+6$ and $11$ in base $r$ is $r+1$ because the leading digit is $r$.  Similarly we have $3\cdot 6 = 22_r=2r+2$  Each of these gives us $r=8$
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