A) We don't tend to 'eat' Chewing Gum, we generally just chew it and then dispose of it. B) Smoking affects you - seriously - no matter what else you may be doing. I don't think that the dangers you face from smoking are lessened, or increased, by chewing gum at the same time. If the gum is a Nicotine Gum, as used by people trying to quit smoking, then you are increasing the dosage of nicotine that your body is getting. Nicotine is highly poisonous.
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No, this would be very highly improbable.
If this refers to "nicotine gum", it is not the nicotine that presents the greatest cancer threat in smoking. However, long-term use of nicotine can have other health problems.
It's harmful to smoke! Chewing, or eating anything opens up the taste buds on the tongue. If you smoke and eat at the same time, you increase the risk of getting cancer of the mouth.
Chewing tobacco (and other "smokeless" tobacco) greatly increases your chance of developing of oral, throat and esophageal cancer.