9. What is the smallest three-digit number $K$ which can be written as $K=a^{b}+b^{a}$, where both $a$ and $b$ are one-digit positive integers?
SolUtion
100

As the problem is symmetrical in $a, b$ we assume $a \leq b$ without loss of generality. If $a=1$ then the maximum value of $a^{b}+b^{a}$ is $1^{9}+9^{1}=1+9=10$. This is not a three-digit number, so cannot be a value of $K$.
If $a=2$ then possible values for $K$ include $2^{9}+9^{2}=512+81=593,2^{8}+8^{2}=256+64=320$, $2^{7}+7^{2}=128+49=177$ and $2^{6}+6^{2}=64+36=100$.
As 100 can be attained then 100 is the smallest three-digit number $K$.