Turing machines and languages -- recursive (enumerable) or not
Hints.


If you're trying to figure out some property of a Turing machine and the only thing you can think of is, "Check every input" then it's probably not recursive.
If a language is RE and so is its complement, both are actually recursive (this is a good way of proving that languages aren't RE: for example, the language of pairs $\langle M,x\rangle$ such that Turing machine&nbsp;$M$ does not halt on input&nbsp;$x$ is not RE for this reason).
If a Turing machine halts within 100 steps, how much of its input has it seen?


Note that L2 and L3 are subtly different so you can't use the same explanation for both (the answers are actually different).