My mom told me, and I bet yours did too, that it takes SEVEN years to digest gum. As a parent, I see the need to occasionally make up fantastical facts, especially if they involve making choking hazards appear less attractive. But there is absolutely no truth to this old wives tale. Sorry, mom, but I caught you in a lie (and only days before Mother's Day).
Although gum is indigestible (which is probably the origin of the 7 year myth), it passes through the human body and is eliminated in the same manner as any other swallowed substance. The sugars in the gum are absorbed by the body in 1 to 2 hours, and the wad of gum leaves the body fairly unchanged, but on a normal schedule.
Interesting factoids: Chewing the resin of trees is an ancient human habit, and the first commercially produced gum, here in the US, was made from spruce resin in 1846, in the fair state of Maine.