Given that the positive integers $x>1$ and $y$ satisfy $2007x-21y=1923$. Find sum of digits of minimum value of $2x+3y$.

Question

Given that the positive integers $x>1$ and $y$ satisfy $2007x-21y=1923$. Find sum of digits of minimum value of $2x+3y$.

Here we have to solve for two variables using only one equation. How is that possible?
I found out that these are called Diophantine Equations but didn't get a clue in solving them.

Answer 1

Divide both sides by 3. $$669x-7y=641$$

Observe $669-7\cdot4=641$, so $(x,y)=(1,4)$ is one solution to the equation.

The general solution would be $(x,y)=(1+7k,4+669k)$ where $k$ is an integer. You can check this by substituting it back into the equation.

Since $x>1$ and we want to minimize $2x+3y$, take $k=1$. $$\implies 2x+3y=2(1+7)+3(4+669)=2035$$

Therefore, the answer: $10$.