### Instruction: Given that the polynomial $x^2-kx+16$ has only positive integer roots, find the average of all distinct possibilities for $k$.### Response: Let's call the two roots $x$ and $y$. So we know $x+y=k$ and $xy=16$.
The factors of 16 are 1, 2, 4, 8, and 16.
And we know that $x$ and $y$ are positive integers.
So if $xy=16$, $(x,y)$ could be $(1,16)$, $(2,8)$, or $(4,4)$.
That's true. And those pairs give us $k$ values of $1+16=17$, $2+8=10$, and $4+4=8$.
And the average of those is $\dfrac{17+10+8}{3}=\dfrac{35}{3}=11\dfrac{2}{3}$.