Push-ups are really part of a calisthenics program that is more predominantly focused more on endurance and flexibility. Push-ups are on the strength building side of the scale for this group of exercises. It is purported that a good routine of calisthenics combined with an aerobic process is one of the most time proven effective fitness programs known. Certainly most of Jack Lalane's exercise regimen consisted of calisthenics and isometric action.
I would consider doing push-ups in conjunction with other calisthenic exercises and other aerobic activity, such as jumping rope. One thing I might add to the routine is pull-ups, which although requiring some apparatus, I still consider within the same gendre.
As far as quantity go, I think Lalane's recommendation is to work it as a circuit until you break a good sweat for at least 20 minutes.
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The ultimate result of shielding men from the effects of folly is to fill the world with fools.
-- Herbert Spencer, English Philosopher (1820-1903)