An architect is designing a house in which a 104 square foot room and a 130 square foot room share a wall what is the greatest possible integer length of the shared wall

Since the area of a room is its length times its width, you'll need to find a common width, which will be the greatest common factor. The greatest common factor of 130 and 140 is 10. So the shared wall can at most be 10 ft. long.