Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 4 / 1E / Solution

The Euclidean algorithm givesThus the greatest common divisor is $d=7$. The extended Euclidean algorithm reverses these divisions: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 $x-8$, givingEvery displayed pair satisfies the original linear Diophantine equation, so the parametrization is exhaustive.

For the modular arithmetic calculation, $21\cdot 18=378\equiv 1(\mathrm{mod}29)$, so $18$ is the modular inverse of $21$. Multiplication gives $n\equiv 25\cdot 18\equiv 15(\mathrm{mod}29)$. The admissible integers are $15+29t$ for $0\le t\le 68$, ending at $1987$; the next is $2016$. There are $\boxed{69}$ such integers.