it depends on the amount aspirated. Small amount may lodge in a distal part of the bronchial tree and may not give symptoms. But large amount will definitely give symptoms. Chest Xray may show if aspirated. thank you
You would probably get a chronic cough. But it's best to be safe. If you felt it go when you breathed in, get an x-ray! It could cause some serious damage in your lung.
I would guess a piece of gum in one's lung would cause symptoms. Best to go in and get a chest x-ray to look to see if the gum is there, though, if it indeed was aspirated, not swallowed.