Themed Challenge 5 (Finn Hulse): Finn-umbers!

First, we have to find the smallest Finnumber. First, consider $ab=10$ . Here, the value of $n$ can be either $1,2,5$ , or $10$ , as $n$ must divide $ab$ . However, it must be noted that $n$ can never be $1$ , as $n$ would always divide $abc$ . Also, $n$ cannot be $2, 4$ , as $bc$ would be divisible by $2$ or $4$ , then so would $abc$ . Also, $n$ cannot be equal to $5$ because $c$ would have to be either $5$ or $0$ , and then $abc$ would be divisible by $n$ . In this case, if $n=10$ , then $bc$ must be $00$ . But, $10$ divides $100$ . Thus, there are no Finnumbers from $100-109$ .

Now, if $ab=11$ , then $n=11$ . Thus, $bc$ must be $11$ , and $abc=111$ , which is not divisible by $11$ . Thus, the smallest Finnumber is $111$ . Thus, to minimize $z$ , $z=111$ . Now to maximize $x$ , we must check if $x$ can be the largest Finnumber. Now, the largest three-digit number is $999$ , which is a Finnumber, with $n=99$ . Now, if $x=999$ , then $y$ must be $888$ . After checking it is found that $888$ is actually a Finnumber, with $n=88$ . Now, that the maximums of $x,y$ are obtained, $x+y=999+888=\boxed{1887}$