Hi David!

Here are some hints:

8.23: Compare with example problem 8.3. Since $60$ divides the product $abc$ for any Pythagorean Triple, $60$ must divide this largest number we are looking for.

8.27: Compare with example problem 8.6. There we found one solution assuming $x$ and $y$ were positive. Can you somehow use this solution to find another solution that may have $x$ or $y$ negative?

8.29: Notice that $f(n) = \dfrac{(n+2)(n-1)}{2}$. For any $n$ we always have that one of $n + 2$ and $n-1$ is even and the other odd. Now, we want that after dividing by $2$ the product of these is still even, so we want to find conditions for $n$ to make sure that either $n+2$ or $n-1$ is a multiple of $4$.  Try to do something similar with $f(n+1)$. Remember the goal is to find values of $n$ that make both $f(n)$ and $f(n+1)$ even at the same time.