 Solution

The extended Euclidean algorithm givesBack-substituting in these equations produces the Bezout identityThus one integer solution is $\boxed{x=12,y=-7}$. In fact, subtracting this Bezout identity from any other solution and using that $17$ and $29$ are coprime integers gives all solutions: $x=12+29t$, $y=-7-17t$, where $t\in \mathbb{Z}$.

For the modular arithmetic calculation, $29$ is a prime number and $17$ is coprime to it, so Fermat little theorem gives $17^{28}\equiv 1(\mathrm{mod}29)$. Since $138=5\cdot 28-2$, and the preceding Bezout identity identifies $12$ as the modular multiplicative inverse of $17$, we obtainConsequently the least positive representative is $\boxed{28}$.