I don't have time to go to the gym. Will doing 100 pushups a day keep me in shape?

I need to stay in shape, but I don’t have time to go to the gym. Would 5 sets up 20 push-ups every day keep me healthy and fit? I’m not trying to get huge muscles or anything, just keep my blood flowing and stay in shape. I would run, but I despise it and get intense cramps – I find push-ups to be more enjoyable.

Would it be enough so that when my doctor asks if I exercise daily, I could tell him yes?