An element has atomic number 10 and an atomic mass of 20. how many neutrons are in the atom of this element?

Atomic mass is found by adding the number of protons and neutrons together. The atomic number tells you how many protons an element has. 
So since this element has 10 protons and an atomic mass of 20, there are 10 neutrons in this element. 

Mass is the amount of matter in an object.  weight is the measurement of the pull of gravity on an object.  say for example, you travel to the moon which has a different gravitational field compared to the gravitational field of earth. (gravitational fields: earth = 1, moon = 0.165)  now when you are on the moon, the pull of gravity on you is different than on the earth. and as weight is the measurement of the pull of gravity on an object, that means yes, your weight can and will change on the moon.  however, on the moon, on earth or even on jupiter, the amount of mass inside of you remains the same. meaning your physical mass remains constant.  in conclusion it is possible for your weight to change as your mass remains constant.