When you swallow gum it is most likely to just pass through your digestive system. The myth that it stays in your stomach started because gum takes seven years to be completely broken down by stomach acid
Dammit! It doesn't? But at the end of the year in 5th grade our teacher gave us all gum and made us swallow it so the week of graduation we'd all finally poop it out!
EDIT: I check the poo not the gum