Finding integer solutions for $6x+15y+20z=1$ Problem: Find integers $x$, $y$, and $z$ that satisfy the equation $6x+15y+20z=1$.
I noticed that $\gcd(6,15)=3$, $\gcd(15,20)=5$, and that $\gcd(6,20)=2$. And of course $\gcd(6,15,2)=1$. 
Of course I know to set one of the variables (x, y, or z) to be zero. But I want to know how to get the more trivial answers. But where do I go from here to find the solutions?

Hint: Using Euclidean Algorithm you can find the gcd of the 3 numbers. Try using Euclidean Algorithm in the inverse direction some way.
gcd(a, b, c) = gcd(gcd(a, b), c)
$15=6 \times 2+3$ so $3= 1 \times 15-6 \times 2$ (+)
$20=6 \times 3+2$
$3=2 \times 1+1$
$1=3-2=3-20+6 \times 3=-20+7 \times 3=-20+7 \times 15-14 \times 6$ using (+)
So $z=-1$,$y=7$,$x=-14$ is a solution
Hope this helps.
