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.