Finding positive Bézout coefficients If I have Bézout coefficients (obtained using extended Euclidean algorithm) and one of them is negative, what is the easiest way to obtain positive Bézout coefficients?

Let $a,b,c$ be natural numbers. Can we solve $ax+by=c$ in naturals in a better way than using the euclidean algorithm? Obviously if $c < a+b$ we can't and if gcd(a,b) doesn't divide c we can't so lets suppose a,b,c coprime then we only really need to check if $y = (c-ax)/b$ is a positive integer for $1 \le x \le (c-b)/a$ as soon as its negative we can stop short knowing there are no solutions.
