Show that one and only one of x,x plus 2 and x plus 4 is divisible by 3?
The remainder when x is divided by 3 is 0,1,2
If 0, we're done
if 1, then x=3n+1, so x+2 = 3n+3. Done
if 2, then x=3n+2, so x+4 = 3n+6. Done
No other choices.
If 0, we're done
if 1, then x=3n+1, so x+2 = 3n+3. Done
if 2, then x=3n+2, so x+4 = 3n+6. Done
No other choices.