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Generally speaking, it is proper for a person to rinse his mouth in the morning before he prays, in order to ensure that he prays to the Almighty with a clean mouth. On a fast day, however, the concern arises that one might inadvertently swallow some water while rinsing, in violation of the fast. Hacham Ovadia Yosef therefore rules that in the morning on a fast day, one should rinse his mouth with less than a Rebi’it (slightly more than 3 oz.) of water, so as to ensure that he does not violate the fast. One should also keep his head bent downward and spit the water out immediately, without gargling, to avoid swallowing water.
One who wishes to brush his teeth on a fast day may do so, provided that he rinses his mouth with less than a Rebi’it of water.
May one chew gum on a fast day?
The authorities debate the question of whether chewing constitutes "eating" in the Halachic sense. Hacham Ovadia Hedaya (1890-1969), in his work Yaskil Abdi, rules that chewing gum is not considered "eating," and thus one does not recite a Beracha before he chews gum. Hacham Ovadia Yosef, however, in his work Yabia Omer (vol. 7, Orah Haim 33), disagrees, and rules that if one chews gum coated with sugar or other sweetening agent, he must recite the Beracha of She’hakol. By the same token, it would be forbidden to chew sweet chewing gum on a fast day. One may, however, chew a substance that has no flavor, such as wax used by people wearing braces, as this certainly does not constitute "eating."
A person suffering from a headache on a fast day may take pain relieving pills. If possible, he should ingest the pills without water. However, Hacham Ovadia Yosef rules that if one cannot swallow pills without water, then he may use a small sip of water, though preferably he should try ingesting the pills without water.
Summary: On a fast day, one may rinse his mouth and brush his teeth, provided that he puts less than 3 oz. of water in his mouth, and takes precautions not to swallow any water. One should not chew gum on a fast day, though one may chew flavorless substances, such as wax used by those wearing braces. One who must take pain relieving pills should ingest the pills without water; if necessary, though, he may use a small sip of water to help him swallow the pill.