I'm nearing sixty and am working my way from being bed-ridden for three years with a damaged lower back to as fit a condition as I can be in. I used to be a "gym-rat," but those days are permanently over. Instead, I just do some door-gym stuff, and push-ups, squats, etc.
Just a bit ago, I finally broke 100 pushups. I do them all in four sets in ten minutes. I'm working to get down to just one set. But I have pushup handles that allow me to dip my chest below the level of my hands as I do the pushup. I also elevate my feet so that my mid and upper chest get a good workout. And, of course, I fully extend on the press up. For a guy my age, I can tell you that these all make a big difference in reaching the 100 mark. Whew!
I have, then, a lot of respect for anyone who can do 100 pushups in five minutes! Well done!