Well, it isn't actually a cactus. Rather, it is some species of Euphorbia. I'm not sure about the species, although chances are it is either E. ammak, E. acrurensis, or maybe even E. trigona. Pruning one of these plants can prove challenging. All Euphorbias produce a thick white latex, that when you cut the plant, bleeds out. The latex is toxic, and usually a potent irritant. Getting any of the latex in your eyes or on mucus membrane will prove exceptionally painful, and in some cases, can cause permanent damage. Therefore, before you embark on a pruning such a large plant, donning some safety glasses, gloves, long sleeves and pants would be a good idea. This warning may seem a little extreme, and usually it is. However, some people are very sensitive while some plants are particularly potent. Given that, it is better to be safe than sorry, and to use the appropriate precautions. With the requisite warning out of the way, you'll be pleased to learn that the actual pruning of these plants is straightforward. As you suspected, cutting at one of the joints is the easiest way to do, and will keep the plant looking more attractive. You can cut right across a stem, but it doesn't look as nice, and will bleed like mad. Keep a rag handy to wipe up the latex that bleeds out. The last bit of advice I might offer, is to wait until spring to do any major pruning. I realize that since this is in an office, the spring won't be any warmer, but there will be more light. Pruning induces branching, and if there isn't much light (like winter in NY), then the new growth will be very leggy and tall, and not very healthy either.

Okay! Thanks for your input. I just couldn't find one that looked close enough to this species so I just wanted to make sure. And actually this picture was after I cut off the bad parts.The stem was pretty much broken off so I just cut off the gross gooey parts. The rotted parts were turning black and have some white fuzz (mold?) on it. I was too grossed out to remember to take a picture xD Do you think it's too small to make it? My friend was about to throw it out and I wanted to try and save it :o

Erica, the plant I have is from a cutting I took from a larger plant I had. When I was given the original plant, the previous owner had already removed a few cuttings, and the scars where they were taken were quite noticeable. I have to say it did bother me a bit. Jeff is right, it is a good plant for taking cuttings from, as it produces multiple growing points, but bear in mind that it will affect the look of the plant. If that doesn't bother you, then prune away. Christopher

Thanks :) Looks like a mini cholla, based off a quick search of opuntia type cactus. Also, based off this search it looks a little spindly, perhaps. It's an ouchy little thing. Should I prune the long arms?