Praise be to Allah.
Washing the arms is one of the obligatory parts of wudoo’, without which it is not complete. That is proven by the Qur’an and Sunnah, and by scholarly consensus.
An-Nawawi (may Allah have mercy on him) said: Washing the arms is obligatory according to the Qur’an and Sunnah, and (scholarly) consensus.
End quote from al-Majmoo‘, 1/418
Ibn Qudaamah (may Allah have mercy on him) said: With regard to washing the arms up to and including the elbows there is no difference of opinion among the scholars of the ummah that it is obligatory to wash the arms when doing wudoo’. Allah, may He be exalted, stated that in the verse (interpretation of the meaning): “wash your faces and your hands (forearms) up to the elbows” [al-Maa’idah 5:6].
End quote from al-Mughni, 1/85
The one who told you that is either ignorant and does not know the simplest thing about Islam or you did not understand what he said properly.
In that case what you must do is learn from this an important lesson in life: that you should not ask anyone about your religion except one who is known to be knowledgeable; you should not ask the first one you meet or learn your religion from anyone who speaks about it, even if they do so without knowledge.
Moreover, you should pay proper attention to what is said to you, so that you will understand it correctly.
If a person omitted one of the conditions, essential parts or obligatory parts of an act of worship because he was unaware of what was required of him, then he is excused because of his ignorance and there is no sin on him for what he did out of ignorance. He is also not to be instructed to repeat the prayers and so on in which he made mistakes.
Based on that, if a person did wudoo’ and did not wash his arms or any of the parts that must be washed in wudoo’, he has to repeat the prayers that he offered, because he did not do wudoo’ as enjoined by Allah, unless he was unaware of the ruling, in which case he does not have to repeat them, especially when there are so many prayers. But what the Muslim must do in the future when he does not know about some matter of his religion is ask someone whose religious commitment and knowledge he trusts, so that he may do what Allah has commanded him to do with proper knowledge and insight.
With regard to rinsing the mouth and nose in wudoo’, there is a significant difference of opinion among the scholars. The majority of scholars are of the view that it is not obligatory to rinse the mouth and nose in wudoo’.
The view on which the fatwas on our website are based is that both actions are obligatory in wudoo’.
This has been discussed in the answer to question no. 153791
But if you consulted some scholars who issued a fatwa saying that it is not obligatory, there is no blame on you for that; therefore one is not required to make up any of the prayers that one offered without rinsing the mouth and nose.
And Allah knows best.