The correct answer is option 3. Here`s why:

Option 1: "Every great circle divides the Earth into equal halves." This is true. By definition, a great circle is any circle that cuts a sphere into two equal hemispheres.

Option 2: "Every great circle is a circumference of the Earth." This statement is also true. Because a great circle divides the sphere into two equal halves, it always represents the total circumference of that sphere or, in this case, the Earth.

Option 3: "Great circles mark the longest travel routes between locations on the Earth`s surface." This is incorrect. Great circles, in fact, represent the shortest possible distance between two points on a sphere. Thus, travel routes along great circles are the most direct, not the longest.

Option 4: "A great circle is the largest circle that can be drawn around the Earth through two particular points." This statement is true. Great circles are indeed the largest possible circles that can be drawn around the Earth, hence the name "great" circles. They cross the Earth`s surface along its diameter, making them the largest possible circles.