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Had Joseph and Mary lived in Nazareth previous to the birth of Jesus?
Luke: They had.
Matthew: They had not.
“And Joseph also went up from Galilee, out of the city of Nazareth, unto the city of David, which is called Bethlehem, ... to be taxed with Mary his espoused wife.... And when they had performed all things according to the law of the Lord, they returned into Galilee, to their own city Nazareth” (Luke ii, 4, 5, 39).
“When he [Joseph] arose, he took the young child and his mother by night, and departed into Egypt: and was there until the death of Herod.... But when Herod was dead, ... he arose, and took the young child and his mother, and came into the land of Israel. And when he heard that Archelaus did reign in the room of his father Herod, he was afraid to go thither; notwithstanding, being warned of God in a dream, he turned aside into the parts of Galilee: and he came and dwelt in a city called Nazareth” (Matthew ii, 14–23).
According to Luke their home was in Nazareth of Galilee; according to Matthew their home was in Bethlehem of Judea. Luke states that they merely visited Bethlehem to be enrolled for taxation and fulfill a certain Messianic prophecy. Matthew states that after the flight into Egypt and the death of Herod they were returning to Judea when fearing Archelaus they turned aside into Galilee to avoid this ruler and fulfill another Messianic prophecy.