Necessary condition of Diophantine equation to have integer solution?

Question

I have answered the questions on the attached image. I am having trouble answered d) which asks for a condition for equation

$$ax+by=c$$ to have integer solutions.

For a) it is quite simple to see that $gcd(4,9)=1$ which allows us to use $ax+by=c$ as $4x+9y=1$, so if we multiply by $7$ we get $28x+63y=7$. For b) my assumption is as $gcd(6,9)=3$ it is actually not possible to achieve this.

Thank you!

Answer 1

The equation $ax+by=c$ has integer solutions iff $gcd(a,b)\mid c$.

Proof: Suppose we have a solution $(x,y)$. Since $gcd(a,b)\mid a,b$, we conclude $gcd(a,b)\mid ax+by=c$. Conversely, suppose $gcd(a,b)\mid c$. We write the greatest common divisor as a linear combination, $gcd(a,b)=ka+lb$, and multiply this equation with the integer (!) $r:=c/gcd(a,b)$ to get a solution, namely $x=rk$ and $y=rl$.

EDIT: Sorry for the double post, I started typing before the comment was posted.