greatest common divisor and solution in integers The greatest common divisor of 203 and 147; $gcd(203,147)=7$. 

Thus how can we find all the solution in integers $x,y$ of the equation $203x + 147y=7$?

First off we can use the Extended Euclidean Algorithm to find some "first" values of x and y.
$$\begin{array}{rclcrcllr}
1 & \cdot & 203 & + & 0 & \cdot & 147 & = & 203 \\
0 & \cdot & 203 & + & 1 & \cdot & 147 & = & 147 \\
1 & \cdot & 203 & - & 1 & \cdot & 147 & = & 56 \\
-2 & \cdot & 203 & + & 3 & \cdot & 147 & = & 35 \\
3 & \cdot & 203 & - & 4 & \cdot & 147 & = & 21 \\
-5 & \cdot & 203 & + & 7 & \cdot & 147 & = & 14 \\
8 & \cdot & 203 & - & 11 & \cdot & 147 & = & 7 \\
-21 & \cdot & 203 & + & 29 & \cdot & 147 & = & 0 \\
\end{array}$$
These last two are important!  The first tells us one solution of the equation; the second tells us a thing that, when I add it in to the first, doesn't change the right side.
So, our result is: $203x+147y=7 \rightarrow (x,y)=(8-21k,-11+29k)$
