Can You Chew Gum while Fasting?
Question

I’ve found that the ruling for chewing gum in Ramadan is that it is not permissible. I would like to know how this conclusion came about and all the alternatives I could use if chewing gum is indeed haram during the month of Ramadan.
Praise be to Allah.Does chewing gum break your fast? 
It seems that the questioner is referring to the kind of chewing gum which is well known nowadays, which is a rubbery substance that usually contains sugar and artificial flavorings. Chewing this kind of gum leads to breaking the fast, because the sugars and flavorings dissolve in the saliva and enter the stomach. 
Undoubtedly this breaks one’s fast, because nutrients enter the stomach. But if the gum does not contain any substance that will dissolve and enter the stomach, then it does not break the fast. 
How to remove the smell of bad breath in Ramadan 
With regard to alternatives to chewing gum, if what is meant is something that may be used to improve the smell of the mouth, then the Muslim should use siwak , because it is one of the Sunnahs narrated from the Prophet (peace and blessings of Allah be upon him). 
You may also rinse your mouth so as to remove the smell , or use toothpaste, subject to the condition that nothing of it reaches your stomach. If you fear that some of it may reach your stomach then you should not use it. 
You should also note that this smell that comes out of the mouth because the stomach is empty and which may disturb a person, cannot be removed with the siwak , etc. because it comes from the stomach due to fasting. It is better before Allah than the fragrance of musk. It was narrated from Abu Hurayrah that the Prophet (peace and blessings of Allah be upon him) said: “The smell of the fasting person’s mouth is better before Allah than the fragrance of musk.” (Narrated by al-Bukhari, 5583; Muslim, 1151)
But if the purpose of using gum is to treat the jaw by means of moving it, please see question no. 38552 . 
And Allah knows best.