Prove that for $n\ge 8$ there are nonnegative integers x and y s.t $3x+5y=n$ 
Prove that for every integer $n\ge 8$ there are nonnegative integers $x$ and $y$ such that $3x+5y=n$ 

Attempt: First of all I want to make it clear whether zero is a nonnegative integer. It should be since it is necessary for the statement to work. Second, I am not sure what method to use. Will proof by cases work? I will have four cases: x is even y is even, x is even y is odd, x is odd y is even, x is odd y is odd. I am not sure how to show that it will cover all integers greater than eight. Thank you for your help.

HINT:
$$3m=3\cdot m+5\cdot0\text{ for }m\ge0$$
$$3m+8=3\cdot (m+1)+5\cdot1\text{ for }m\ge-1$$
$$3m+7=3(m-1)+5\cdot2\text{ for }m\ge1$$
