Your basic issue is that you don't understand switch statements. A switch statement is not a case. To do it with a case statement you'd need to do this:
Console.WriteLine("your input is zero.");
Console.WriteLine("your input is even.");
I agree, this is not a good place to use a case statement, but this is almost certainly what your teacher wants to see. The important points:
1 - a case statement means you list the cases you want to catch, a case is NOT a place to write evaluative code, as you did
2 - there is no reason to use goto here, or, basically, almost anywhere. I have never used goto in my code. It makes your code unreadable and is unnecessary.
3 - you can put more than one case in a block, so I did 2,4,6,8, etc, then put the one block of code that executes for all those cases.
I should add a couple of things:
1 - we don't normally do people's homework. I was happy to help you b/c you were trying to do your own work, not just asking for a solution, but it does seem to me like you were struggling a bit with the way case statements work, and I'd recommend you also talk to your teacher about this, to make sure it's clear in your mind.
2 - you should not push the 'answer' button to ask more questions. You should edit your post ( I copied your replies into your main post ), or use the forum under the answers section.