What the Army calls a push-up and what civilians call a push-up is often very different. It is very rare to find a soldier who regularly does 100 push-ups to standard; I know of only one person who can do it. I did 61 the other day which is pretty good for a former obese 38 year old. BTW, we only have 2 minutes to complete this; resting in the authorized position (downward dog) is allowed.
If your serious about going to the Rangers you need to be training for RASP:
Found most of the advice present solid, to include that doing the exercises you're training to as well as working the main and supporting muscles for an exercise will help. No "magic workouts" here to make X easier.
Remember that technique is equally or more important than the muscles required to do the exercise. As Marines we don't do pushups, our PT test 'roadblock' is typically the pull ups, and form is absolutely essential unless you're a genetic specimen. Chin over the bar means chin over the bar, not neck, shoulders, or chest (which more often than not you absolutely see young Marines doing during the test).
There's also the mental component. If you've never done 20 pull ups then it seems damn near insurmountable. But once you get your twenty, you're going to continue to achieve that number more often than not based on pride alone (with intermittent workouts of course).
Personally, I despise running and "working my abs" to get those 100 crunches and required run time for our physical fitness test, and when I'm destroyed with work week after week my run time shows. I'm beginning to hit the age in my life (30) where I can't just go out there a week or two before the test and run out the miles and get the scores I used to.
Basically, the moral of this long-winded post is "train smarter, not harder" which seems to be a recurring theme amongst Mark's posts (work out as minimal as possible and get results so he can continue to go do fun stuff with his life). Adapting a primal/paleo lifestyle isn't the easiest to do in the military, especially as an young infantryman but neither is it impossible.
I'm happy to share my experiences with you as well if you'd like, merely send me a message and we can talk. Again, I'm a Marine Infantryman and not an Army Ranger, but you'd be surprised how much of our lifestyles transfer over.