$1)$ You have a grating with $500$ lines per mm$.$ You shine a $650$ nm $($red$)$ laser on it$.$ How many diffraction maxima do you expect to see? Show your calculation.

$2)$ What is the angle for the m$=1$ diffraction maximum? Show your calculation.

$3)$ You now shine a $420$ nm $($blue$)$ laser on the same grating. How many diffraction maxima do you expect to see? Show your calculation.

$4)$ What is the angle for the m$=2$ diffraction maximum? Show your calculation.