The most plausible statement is:
The object started on Earth, was transported to space, and was deposited on the moon.
This is because weight is dependent on the gravitational force acting on an object, which varies depending on the location of the object (Earth, space, Moon). While the mass of the object remains constant, its weight will change in different gravitational fields.
Here's a brief overview of the other options:
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The object gains matter and then loses it. - This would change the mass of the object, leading to a weight change, but the scenario states the same mass is maintained.
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The object started at the bottom of a hill, then moved to the top of the hill, and then moved back down to the bottom of the hill. - In this case, the weight of the object remains the same throughout; only its position changes relative to Earth's surface.
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The object changes shape, which changes its weight. - Changing shape does not affect weight as weight is dependent on mass and gravity, not shape.
Thus, the statement about being transported from Earth to space to the Moon accurately describes a situation where the same mass has different weights.