Is there a simpler proof that $f(x,y,z) = 4x + 11y + 18z$ is surjective?

Question

So, one could use $f(x,y,0) = 4x + 11y$. Determining if that function is surjective on $\mathbb{Z}$ takes proving that $4x + 11y = c$ has a solution for every $c$.

Since we know about the existence of the solution to the Diophantine equation $a x + b y = c$ in case $c$ divides $\gcd(a,b)$. We can conclude, since $\gcd(4,11)=1$ that the function is surjective.

Although, let's assume we do not know about Diophantine equations, how would one prove the surjectivity of the function above?

Answer 1

Go ahead and find a solution for every $c$. A simple way is to find a solution for 1, then multiply it by $c$.

Answer 2

Alternatively, notice that the range of $f$ is the ideal generated by $4$, $11$, and $18$, since $\Bbb Z$ is a PID we know

$$\langle \gcd(4,11,18)\rangle = \langle 4,11,18\rangle$$ and $\gcd(4,11,18) = 1$ so $\langle\gcd(4,11,18)\rangle=\Bbb Z$.