Show that any integer $n>7$ can be written as the sum of $3$s and $5$s exclusively 
Show that any integer $n>7$ can be written as the sum of $3$s and $5$s exclusively, 

i.e., 
$$
8= 5+3 \\
9=3+3+3  \\
10 = 5+5 \\
11 = 5+3+3 \\
12 = 3+3+3+3 
$$
So I've started in a couple directions without progress.  I think it makes sense to write
$$n=3x+5y$$
for $x,y \geq 0$, observing that either $3\mid n$ or $5\mid n$.        
It also seems that if you are at $n$, you can increment to $n+1$ by replacing a $5$ with two $3$s.  Or you can take $n-1$ and replace a $3$ with a $5$.  In this way, it seems like you should be able to keep incrementing after $n=8$, but these ideas aren't formalizing into anything...

Since $8=5+3$ we consider those $n$'s for which $n\ge 9$.


*

*If $n=3k$ then we are done. 

*If $n=3k+1$ then observe that $3k+1=3(k-3)+2\cdot 5$.

*If $n=3k+2$ then observe that $n=3k+2=3(k-1)+ 5$. 
