Once the elbow is removed the rest of the downspout should slide right out.
How to remove seamless gutters.
Now it is usually slipped completely out without damaging the gutter.
If the gutter is very long be cautious removing it.
Put the claw of 1 hammer on the head of the gutter nail then hit it with the other hammer when the nail is loose give it a firm follow through hit.
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.
With the above safety procedures in mind use a power drill in reverse or a screwdriver to remove the screws underneath the roof or in the siding holding the gutter in place.
Then you can easily remove the straps utilizing the screwdriver drill or hammer and pry bar depending on what was used to attach the straps or supports to the house.
To remove the downspout you will have to remove the elbow at the bottom of the downspout.
This is usually done by removing screws or using a pry bar to remove nails.