Call the integers $ n-1 $, $ n $, and $ n+1 $. Their mean is $ n $; their product is $ (n-1)(n)(n+1)=120 $ and their product divided by their mean is $ (n-1)(n+1)=24 $. Dividing the first equation by the second, we get $ n=5 $. The largest of the three is $ n+1=\boxed{6} $.