Question

Find the general solution and the least positive integral solution of $455x-519y=1$.

Answer 1

$455x-519y=1$

$-64y+455(x-y)=1$

$⟹-64y+455z=1$ where $z=x-y$

$⟹-64y+(64\times 72+7z)=1$

$⟹72-64(y-7z)=1$

$⟹7z-64t=1$ where $t=y-7z$

$⟹7z-(7\times 9t+t)=1$

$⟹-t+7(z-9t)=1$

$-t+7u=1$ where $u=z-9t$

$thenu=0,$ $\therefore t=-1$

$u=z-9t$ gives $z=-9$

$t=y-7z$ gives $y=-64$

$z=x-y$ gives $x=-73$

$\therefore$ Solutions are given by

$x=-73+519k$

$y=-64+455k,$ where $k=1,$ the least positive integral solution is

$x=446,y=391$