Simply put. Yes.
See the answer below to get an idea as to how they left and returned.
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800 times.
The creator; the God of the Old Testament; the God of Abraham, Isaac and Jacob - read in the Bible the first book, Genesis, which explains the beginnings of everything including the people who descended from Jacob, whose name was changed by God to Israel. The Jews descended from his son Judah.
Very unlikely. The only potentially useful test would be DNA analyses. It is far more likely that the current populations of Arabs in Israel, Lebanon, Syria, Jordan, and Iraq, are, at least partially, descended from the Lost Tribes of Israel.
Abraham's grandson, Jacob, had 12 sons from whom are descended the 12 tribes of Israel.
In the Old Testament, God did not divorce Israel but rather criticized Israel for its unfaithfulness and idolatry. While there are references to God's relationship with Israel being broken due to their sinfulness, the concept of a formal divorce is not explicitly mentioned in the Bible.
No. It is not a requirement or even considered whether or not the Prime Minister of Israel is of Davidic Lineage. Israel is not considered the Messianic country foretold in the Old Testament where the rulers would have Davidic Lineage. Benyamin Netanyahu, the current Prime Minister, has never claimed to be of the Davidic Lineage nor is he believed to be.
King David is in the old testament.
12 sons of Jacob
According to Mat 1:3 and Luke 3:33-34 and Jesus is descended from Jacob's son Judah. "Abraham was the father of Isaac, Isaac the father of Jacob, Jacob [also called Israel] the father of Judah and his brothers, Judah the father of Perez and Zerah" [Mat 1:2-3]
The current president of the State of Israel is Shimon Peres.
Israel