How do you solve an inequality with the factorial of a variable?
Example: Determine the interval of $n \in \Bbb N$ for which the following inequality holds:
$$n! \leq 157788 \cdot 10^{10} $$
Can this be solved using algebraic techniques?
If not, what calculus techniques can I use to solve this inequality?
If you want to think about it algebraically, you would probably try to factor. So, notice the 10^10. 10s can only come from a combination of evens and multiples of 5. (namely, you will be limited by multiples of 5, as there are fewer of them). So, 10^10 implies that you have 10 multiples of 5, but not 11. Thinking about this, however, we note that 50! is way too big, so we have to be more rudimentary.
Thought of a different way, our answer is on the scale of $10^{16}$. This means that it is about 15 things the size of 10 multiplied together. namely 2*5, 3*4,6*7,8,9, 10, etc. to get 15 places this way, we'd likely have to go up to about 16ish. Being realistic, we could try some numbers in this area.
That gives us a good place to start, so we play around in that area, we find that it has to be 17!. (the comment below was absolutely right, I messed up and said 15!)