1. One of his seven brothers is in Malaysia, and the others (all the rest) are in Singapore.
Is it correct to use "the others" in the above sentence? Yes.
2. One of his seven brothers is in Malaysia, and others are in Singapore.
What does "others" mean in the above sentence? It means 'some of the other brothers (we don't know how many)'. Is it possible if we use "other"? No.