A base-10 integer n can be represented $32_a$ as in one base and $13_b$ in another base, where a and b are any integer bases larger than 3. What is the smallest possible sum a+b?
By assumption $a \ge4$ and $b\ge4$, and the assumption $32_a = 13_b$ gives the constraint $3a+2=b+3$, that is, $a = (b+1)/3$. Solving $a=(b+1)/3 \ge 4$ gives $b\ge11$. When $b=11$ we have $a=(11+1)/3=4$. So the smallest possible sum $a+b=4+11=15$.
