Yes, because the changes in how much you see of the moon depend on its position in its orbit around the Earth. This complete orbit takes 29.5 days, and at one point of this cycle it is in the full moon phase (this depends on the relative position of Earth, Moon and Sun). Compared to this, the day/night cycle on Earth is 24 hours due to the rotation of the Earth’s axis. During the day/night cycle the moon will have been visible from every part of the Earth, so everyone on Earth will have seen the full moon. Have I answered this clearly enough?
As Blanka said, the moon moves around the earth once ever 29.5 days. Where it is in relation to the sun and the earth is what makes it full moon, half moon, new moon etc.
Because the earth itself is spinning on its own axis once every 24 hours everybody on earth gets to see the full moon within these 24 hours
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