Here s a quick fix for that problem demonstrate.
How to take off rain gutters.
Remove the gutter once all hangers are gone.
It might sound convenient but letting a large piece of gutter fall to the ground is unsafe.
This is usually done by removing screws or using a pry bar to remove nails.
If the gutter is very long be cautious removing it.
You can make a pry block from scrap 2 4 or 2 6.
This is as simple as removing each gutter hanger or other device that holds the gutter in place.
Then simply place the block in the gutter and use your hammer or pry bar against it to pull out the nail and remove the ferrule.
To remove the downspout you will have to remove the elbow at the bottom of the downspout.
You can use caulk to fill holes left in the siding once the gutters and downspouts are removed.
Gutter nails or spikes used to hold rain gutters in place can work themselves loose with seasonal movement.
This will make removing the rest of the gutter easier.
Your next step will be to remove the downspout.
How to remove a piece of gutter.