= Solution
The <Euclidean algorithm> gives
$$
203=147+56,\quad147=2\cdot56+35,\quad56=35+21,\quad35=21+14,\quad21=14+7,\quad14=2\cdot7.
$$
Thus the <greatest common divisor> is \b[$d=7$]. The <extended Euclidean algorithm> reverses these divisions:
$$
7=21-14=2\cdot21-35=2\cdot56-3\cdot35=8\cdot56-3\cdot147=8\cdot203-11\cdot147.
$$
This supplies the <Bezout identity> with $x=8$, $y=-11$.
To obtain <all solutions of a linear Diophantine equation>, subtract this particular solution from any other and divide by $7$. The result is $29(x-8)+21(y+11)=0$. Since $29$ and $21$ are <coprime integers>, $21$ divides $x-8$, giving
$$
\boxed{x=8+21t,\qquad y=-11-29t,\qquad t\in\mathbb Z.}
$$
Every displayed pair satisfies the original <linear Diophantine equation>, so the parametrization is exhaustive.
For the <modular arithmetic> calculation, $21\cdot18=378\equiv1\pmod{29}$, so $18$ is the <modular inverse> of $21$. Multiplication gives $n\equiv25\cdot18\equiv15\pmod{29}$. The admissible <integers> are $15+29t$ for $0\leq t\leq68$, ending at $1987$; the next is $2016$. \b[There are $\boxed{69}$ such integers.]
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