Question

Find the general solution in positive integers of $29x-42y=5$.

Answer 1

In converting $\frac{42}{29}$ into a continued fraction the convergent just before $\frac{42}{29}$ is $\frac{13}{9}$; we have therefore
$29\times 13-42\times 9=-1$;
$\therefore 29\times 65-42\times 45=-5$;
combining this with the given equation, we obtain
$29(x+65)=42(y+45)$;
$\therefore \frac{x+65}{42}=\frac{y+45}{29}=t$, an integer;
hence the general solution is
$x=42t-65$, $y=29t-45$.