According to Live Science, if the Earth didn't have an atmosphere, we would actually be able to see the moon all month long, because it is out during the day every day except during its full moon phase. The Earth's atmosphere scatters blue light from the sun across the sky, giving it a light blue shade. When it does this, the brightness of that sunlight prevents us from seeing into space, making stars and planets disappear from our perspective.
Despite the Earth's atmosphere blocking most light from space other than the sun's, the stars, planets, and the moon are all still there. During a new moon, we cannot see the moon in daylight because of the brightness of the sky. The moon is still here, but the half that is facing the sun is not the half that is facing us at that time of the month. The phase only lasts around two or three days, so it is only a matter of time before the moon is visible to us again (via Sciencing).
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