FLT from Mochizuki's proof of abc  If Mochizuki's proof of abc is correct, why would this provide a new proof of FLT?
Edit: In proof of asymptotic FLT, does Mochizuki claim a specific value of n and if so what is this value?
 A: Let us suppose that $x^{n}+y^{n}= z^{n}$ with $x, y,$ and $z$ relatively prime. By the abc conjecture, $|x^{n}|\ll |xyz|^{1+\epsilon}$, $|y^{n}|\ll |xyz|^{1+\epsilon}$, and $|z^{n}|\ll |xyz|^{1+\epsilon}$. Therefore, $|xyz|^{n}\ll |xyz|^{3+\epsilon}$ which implies that, for $|xyz|>1$, $n$ is bounded. Unfortunately, this establishes only an asymptotic version of FLT. Nevertheless, if we had explicit information regarding the implied constant in the abc conjecture, we could in principle determine explicit upper bounds for those $n$'s for which the abc conjecture doesn't settle FLT.
UPDATE (October 26th, 2021). In the abstract of "Explicit Estimates in Inter-universal Teichmuller Theory", Mochizuki et. al. we read this:

In the final paper of a series of papers concerning inter-universal Teichmüller theory, Mochizuki verified various numerically
non-effective versions of the Vojta, ABC, and Szpiro Conjectures over
number fields. In the present paper, we obtain various numerically effective versions of Mochizuki’s results... These numerically effective versions imply effective diophantine results such as an effective
version of the ABC inequality over mono-complex number fields [i.e.,
the rational number field or an imaginary quadratic field] and effective
versions of conjectures of Szpiro. We also obtain an explicit estimate
concerning “Fermat’s Last Theorem” (FLT)—i.e., to the effect that
FLT holds for prime exponents greater than $1.615\cdot 10^{14}$—which is sufficient to give an alternative proof of the first case of FLT. In the second case of FLT, if one combines the techniques of the present paper with a recent estimate due to Mihăilescu, then the lower bound $1.615 \cdot 10^{14}$ can be improved to $3.35\cdot 10^{9}$.

If I understand correctly, this paper was uploaded to Mochizuki's homepage four months ago. I dared to update my reply because I consider that the lines above give a definite answer to the OP's second question (regardless of what the generalized opinion about IUTT is).
A: The two answers of J.H.S and Igor Rivin don't really answer the question : they prove FLT
only for an exposant $n$ large enough. 
Since Mochizuki's version of ABC is effective, one could certainly find an effective bound for $n$, and maybe prove by hand the remaining $n$. But as they stand, they are incomplete.
Here is a short proof that Mochizuku implies Fermat. If the proof is correct, since it is also incorrect (cf. the second answer to [this question] by Vasselin Dimitrov), it implies trivially Fermat't last theorem as well as its contrary.
I guess I should explicit my point: it is prematurate to ask precise questions
on the consequence of Mochizuki's proof. As for vague philosophical question, as the one
given in the link above, that have a problem too : they are vague. Let us at least wait for the MO community to accept or not Vasselin Dimitrov's very interesting answer, which
moreover is exactly about the effective version of ABC conjecture needed to answer the PO's question, before asking such questions.
[1] Philosophy behind Mochizuki's work on the ABC conjecture
A: See http://mathworld.wolfram.com/abcConjecture.html
