Here is a link that will provide you with answers. There are 55 peaks and the goal of many weekend climbers is to ascend all of them. For some, it may take as long as 20 years, doing several or more every summer.
http://climbing.about.com/od/mountainclimbing/a/Fourteeners1.htm
Living in NM in the 80’s and 90’s I hiked and climbed several high peaks, but NM does not have any 14,000 feet. The one fourteener I was going to climb in CO was a relatively short hike from a four-wheel drive road. Along with friends, I had my black lab/golden retriever cross with me which went well until the last hundred feet or so which required a rock scramble. He put up such a loud ruckus when I left him behind and started to scramble that I turned back and waited down below for the others to return.
Thank you for the link!