= Solution
The <extended Euclidean algorithm> gives
$$
29=17+12,\qquad17=12+5,\qquad12=2\cdot5+2,\qquad5=2\cdot2+1.
$$
Back-substituting in these equations produces the <Bezout identity>
$$
1=5-2(12-2\cdot5)=5\cdot5-2\cdot12
=5\cdot17-7\cdot12=12\cdot17-7\cdot29.
$$
Thus \b[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 $17^{28}\equiv1\pmod{29}$. Since $138=5\cdot28-2$, and the preceding <Bezout identity> identifies $12$ as the <modular multiplicative inverse> of $17$, we obtain
$$
17^{138}\equiv17^{-2}\equiv12^2=144\equiv28\pmod{29}.
$$
Consequently \b[the least positive representative] is $\boxed{28}$.
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