Greatest integer solution to a factorial inequality?

Questions
1

I’m trying to find the greatest integer solution to a factorial inequality such as:

x! <= 100

x = 4

But I’ve had surprising difficulty finding the functions to do this. I can get a list of every integer solution, but not select the greatest one from the generated list using the commands I found.

2

In[2]:= res = Solve[x! <= 100, x, Integers]

Out[2]= {{x -> 0}, {x -> 1}, {x -> 2}, {x -> 3}, {x -> 4}}

In[3]:= Max[x /. res]

Out[3]= 4
3

Thank you. I tried something similar to that without luck.

4

Just for fun, another possibility is to use InverseFunction:

Floor@InverseFunction[Factorial][100]

(* 4 *)