\[gcd(143,17)=143x+17y\]
where
\[x, \; y\]
are integers, use The Euclidean Algorithm.\[143=8 \times 17+7\]
\[17=2 \times 7+3\]
\[7=2 \times 3+1\]
\[143=8 \times 17+7\]
\[3=3 \times 1+0\]
Hence
\[\begin{equation} \begin{aligned} gcd(143.17) &= 1 \\ &= 7-2 \times 3 \\ &= 7-2 (17-2 \times 7) \\ &= -2 \times 17+5 \times 7 \\ &= -2 \times 17+5(143-8 \times 17) \\ &= 5 \times 143 -42 \times 17 \end{aligned} \end{equation}\]
One solution is
\[x=5, \; y=-42\]
.The general solution is
\[5-17k, \; y=y=-43+143k\]
.