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.

The numbers $3$ and $5$ are realively primes therefore there exists (by Euclid algorithm)  $a,b$ such that $3 a+5 b=1.$ It impliles that $3 an+5 bn=n.$ Now put $x=a n$ and $y= b n.$
Easy to see that $a=2$ and $b=-1$. Thus $x=2n$ $y=-n$.
