What is the reason for shaving pubic hair and armpit hair? What did they use to shave it with at the time of the Prophet (blessings and peace of Allah be upon him)? Is it sufficient just to cut it or must it be shaved?

Praise
be to Allah

Firstly:

The Sunnah indicates that it is prescribed to remove pubic
hair and armpit hair. Al-Bukhaari (5889) and Muslim (257) narrated from Abu
Hurayrah (may Allah be pleased with him) that the Prophet (blessings and
peace of Allah be upon him) said: “The fitrah is five things – or five
things are part of the fitrah – circumcision, shaving the pubes, cutting the
nails, plucking the armpit hairs, and trimming the moustache.”

The wisdom behind the prescription of removing the hair from
these two places – and Allah knows best – is that removing it helps one to
attain a perfect level of cleanliness and prevents what could emanate from
them of bad smells if the hair was left without removing it. And there are
other reasons and wisdom behind it.

Al-Haafiz Ibn Hajar (may Allah have mercy on him) said: These
characteristics of the fitrah may serve some religious or worldly interests,
that one may notice by reflecting upon the issue, such as: improving one’s
physical well-being; cleansing the body thoroughly; taking precautions to
ensure purity when doing ghusl or wudoo’; doing a favour to one’s friends
and companions by avoiding unpleasant smells that may offend them; differing
from the practices of the disbelievers such as the Magians, Jews, Christians
and idol-worshippers; obeying the command of the Lawgiver; preserving that
which is mentioned to in the verse in which Allah, may He be exalted, said
(interpretation of the meaning), “and [Allah] has given you shape and
made your shapes good (looking)” [Ghaafir 40:64]
, because by doing so
one is preserving that beautiful image – it is as if the verse implies: I
have given you beautiful shapes, so do not distort them with anything that
may make them ugly, and take care of them so that they will continue to be
beautiful, for taking care of them is a kind of adhering to dignity and
maintaining harmony with others, because if a person appears handsome or
beautiful, that makes others feel at ease with him, so people will listen to
what he says and appreciate what he says, and vice versa.

End quote from Fath al-Baari.

Secondly:

What was well known at the time of the Prophet (blessings and
peace of Allah be upon him) was that they would use a razor to shave their
hair.

Al-Bukhaari (5079) and Muslim (715) narrated that Jaabir ibn
‘Abdullah (may Allah be pleased with him) said: We were with the Messenger
of Allah (blessings and peace of Allah be upon him) on campaign, and when we
approached Madinah, we wanted to enter the city straight away, but the
Prophet (blessings and peace of Allah be upon him) said: “Delay it until we
enter at night, so that the one who is dishevelled may tidy herself up and
the one whose husband is absent may shave her pubic hair.”

Al-Haafiz Ibn Hajar (may Allah have mercy on him) said in
Fath al-Baari
: This refers to the woman whose husband is absent. What is
meant is so that she may remove her pubic hair, and the word used in this
hadith refers to shaving, because that is the usual method of removing hair,
but that does not mean that it is not allowed to use something other than a
razor. End quote.

Al-Bukhaari (3989) narrated the story of Khubayb ibn ‘Adiyy
(may Allah be pleased with him), in which it says:… when they [the
disbelievers who had captured him] decided to kill him, he asked to borrow a
razor from one of the daughters of al-Haarith so that he could shave his
pubic hair, and she lent it to him…

It says in Musnad al-Imam Ahmad (26705), in the hadith
of Ma‘mar ibn ‘Abdullah (may Allah be pleased with him): … When the
Messenger of Allah (blessings and peace of Allah be upon him) had
slaughtered his sacrifice in Mina, he instructed me to shave his head. So I
picked up the razor and stood by his head. The Messenger of Allah (blessings
and peace of Allah be upon him) looked me in the eye and said to me: “O
Ma‘mar, the Messenger of Allah (blessings and peace of Allah be upon him)
has let you take hold of his earlobe when you have a razor in your hand…”

Thirdly:

The Sunnah with regard to the removal of pubic hair is to
shave it. In the case of armpit hair, the Sunnah is to pluck it, but if a
person simply trims it, there is nothing wrong with that, but it is not what
is best.

Ibn Qudaamah (may Allah be pleased with him) said: Shaving
the pubic hair is mustahabb and is part of the fitrah, and it is very
inappropriate to leave it, so it is recommended to remove it, and whatever
means one uses, there is nothing wrong with it, because the point is to
remove it. It was said to Abu ‘Abdullah (i.e., Imam Ahmad): Is it acceptable
for a man to cut his pubic hair with scissors, even if he does not cut all
of it? He said: I hope it is acceptable, in sha Allah.

End quote from al-Mughni (1/65).

An-Nawawi (may Allah have mercy on him) said: Shaving the
pubic hair is called istihdaad in Arabic because iron (hadeed) is used,
i.e., the razor. That is Sunnah, and the purpose behind it is to cleanse the
area. The best is to shave it, but it is permissible to cut the hair, pluck
it or remove it with a depilatory paste…. With regard to plucking the armpit
hair, that is Sunnah according to consensus. The best is to pluck it for the
one who can stand it, but it may also be done by shaving or using a
depilatory paste. It was narrated that Yoonus ibn ‘Abd al-A‘laa said: I
entered upon ash-Shaafa‘i (may Allah have mercy on him), and the barber was
with him, shaving his armpits. Ash-Shaafa‘i said: I know that the Sunnah is
to pluck it, but I cannot stand the pain.

End quote from Sharh Muslim by an-Nawawi (3/149)

And Allah knows best.