For the ToM I already had 22 fragments that I had already picked up.
I used Destiny tracker to map what I did and didn't have and knocked it up to 37 with maybe an hours effort. Got to go into a couple of strikes and get a few missing tonight, do undying mind and a Phobos mission...job done.
From what I can gather neither mission is a pushover but with a bit of effort and maybe an extra player it should be doable without much fuss.
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Neither mission is bad, just a little challenging. I had one buddy and it was over pretty quickly. I think both targets spawn when the boss is down to a quarter or so.