I got this question wrong on a pop quiz and I need help understanding why.

You and your friend, who weighs the same as you, want to go to the top of the Eiffel Tower. Your friend takes the elevator straight up. You decide to walk the spiral stairway, taking longer to do so. Compare the gravitational potential energy of you and your friend, after you both reach the top.

I selected: Your gravitational potential energy is greater than that of your friend because you traveled a greater distance in getting to the top.

My teacher says: both of us have the same amount of gravitational potential energy at the top.

How? Why?