Prove that for every n ∈ Z, there exist a, b ∈ Z such that $n = 5a + 2b$.

You have $5-(2)(2)=1$, thus $n=5n-2(2n)$ take $a=n, b=-2n$.

Bezout's identity says that there exist integers $x,y$ such that $2x+5y=1$, since $(2,5)=1$. Then just multiply by $n$.

a=1 will work for all odd numbers. :)