Assuming the Earth is a perfect sphere of radius R. By how much would your mass (m), as given by a scale, change if you measured it on the north pole and on the equator?

The key observation here is that the Earth is spinning (angular velocity w) and so are you. The scale will give one number or another depending on the force that you exert on it, and by Newton's 3rd Law that is equal and opposite to the force that it exerts on you (i.e the normal force). On the north pole you are sitting just on the axis of rotation, so the centripetal force is zero. However, on the equator the centripetal force is no longer zero, so the normal has to be slightly smaller than your weight to keep you rotating. Bringing in some maths: Centripetal force= Your weight - normal N=mg-mRw^2=mg(1-rw^2/g)= what the scale "thinks" you weight. Hence, the readings are different by a factor of (1-rw^2/g)