1. Mr. Gates has four sons. One lives in Chicago, and the others live in L.A.
2. Mr. Gates has four sons. One lives in Chicago, and the rest live in L.A.
3. Mr. Gates has four sons. One lives in Chicago, and the rest lives in L.A.
(Which ones are correct? are they the same in meaning?)
1 and 2 are correct, and yes, they mean the same thing.
3 is incorrect because "rest" is plural, and it needs the verb to be "live" as in sentence 2.