Answer:
You'd really have to define what you mean by a "lunar month".
The moon completes an orbital revolution of the Earth every 27.32 days.
But a complete cycle of the moon's "phases" that we see takes 29.53 days.
If you lived on the moon, here's what you would see in your sky:
-- The Earth would never move in your sky. Wherever you see it right now is where
you'll always see it. Of course, that means that there is half of the moon's surface
from which the Earth is never seen.
-- While it never moves in the sky, the Earth does go through 'phases' ... New Earth,
waxing crescent Earth, First Quarter, gibbous Earth, Full Earth, etc.
-- Although the Earth never moves in your sky, the Sun does. It rises, crosses the sky,
and sets. It's up for roughly 2 earth-weeks, then down for roughly the next 2 earth-weeks.
-- You would soon notice that the 'phases' of the Earth exactly track the changing
position of the sun in your sky. The Earth stays right where it is. The sun goes
round and round, rising and setting. The amount of Earth that's illuminated
exactly tracks how far the sun is from the Earth in your sky.
-- If you sit outside with a glass of iced tea and watch the Earth, you can clearly
see it spinning, as the continents and cloud-covered areas move across it. You can
see it make a complete spin about once every 24hours 50minutes . But it's slipping
around inside this partly-illuminated shape of the current phase, which isn't changing
fast enough to notice at all.
-- The farther the sun is from the Earth in your sky, the more of the Earth is lit up.
When the sun is up, the Earth phase shrinks from Third Quarter, through waning
crescent, to New Earth, and then grows through waxing crescent, to First Quarter
when the sun sets.
While the sun is down, the phase goes from First Quarter, through waxing gibbous
to Full Earth, then shrinks through waning gibbous, to Third Quarter when the sun
rises.
-- This whole cycle ... from one sunrise to the next, through all the phases of the Earth,
runs 29.53 earth-days.
-- Because there is no air to scatter the sunlight or spoil your view, you can see
the stars in your sky all the time, even when the sun is up. So you also get to
know the constellations and their motions very well. You notice that the whole
dome of stars rotates around you, a little bit faster than the sun does. If you
watch one star, you observe that it rises and sets, just like the sun does, but
a little faster than the sun. The sun takes 29.53 earth-days, but the whole bowl
of stars rotates around you every 27.32 earth-days.
-- So the sun seems to be slowly losing ground, drifting 'backward' through the stars ...
the sun takes 2.21 earth-days longer to go all the way around you than the stars
do, and the stars are continually pulling ahead of the sun. How long does it take
the stars to completely 'lap' the sun in your sky ? ... make one more complete
revolution than the sun has made in the same amount of time ?
Watch it for several years so you can average it out, and it turns out to be 365.25 earth-days.
Where have you heard that number before ?