Number theory GCD word problem 
An oil company has a contract to deliver 100,000 litres of gasoline. Their tankers can carry 2,400 litres, and they can attach one trailer carrying 2,200 litres to each tanker. All the tankers and trailers must be completely full on this contract, otherwise, the gas would slosh around too much when going over some rough roads. Find the least number of tankers required to fulfill the contract. Each trailer, if used, must be pulled by a full tanker.

What I can gather from this is the following equation:
$$100000=2200x+2400y,\ x\leq y$$
What I know about the GCD is that for two integers $a,b$ with at least one of them non-zero, $$g=ax+by$$ is the "least positive" $bx+cy$ where $x,y\in\mathbb{Z}$.
The problem I have is that I think I'm doing something wrong... $(2400,2200)=200(11,12)=200$. So from this I gather $$200=2200x+2400y$$, but this implies that $x=-1$ and $y=1$. $$100000=500(2200x+2400y)$$
This doesn't help either as I still get a positive number and a negative number.
Am I misunderstanding the question? What am I doing wrong?
 A: Actually you don't do anything wrong, but you need just to complete the solution. We have:
$$2200(-1) + 2400(1) = 200$$
The number in the parenthesis are the solution for $x$ and $y$ respectively.
Now we have:
$$500(2200(-1) + 2400(1)) = 500 \cdot 200$$
$$2200(-500) + 2400(500) = 100000$$
But this equation has infinitely many solution so we generate them. We can add $\frac{2400}{200} = 12$ to the value of $x$ and then subtract $\frac{2200}{200} = 11$ of the value of $y$, and we'll generate a new solution. Because we can do this infinite amount of times, we can write it as:
$$2200(-500 + 12n) + 2400 (500 - 11n) = 100000$$
In our best interest is to choose $n=42$, so we have:
$$2200(-500 + 12n) + 2400 (500 - 11n) = 100000$$
$$2200(-500 + 12\cdots 44) + 2400 (500 - 11\cdots 44) = 100000$$
$$2200(4) + 2400(38) =100000$$
So the least amount of tankers is $4+38=42$. Note that if we chose $n=41$, we'll have negative number of trailers.. Note that also $n=43$ works:
$$2200(16) + 2400(27) =100000$$
But that's $16+27=43$.
