Question

Find the general solution and the least positive integral solution of $775x-711y=1$.

Answer 1

$775x-711y=1$

$(711x+64x)-711y=1$

$⟹64x+711z=1$ where $z=x-y$

$⟹64x+(64\times 11z+7z)=1$

$⟹7z+64(x+11z)=1$

$⟹7z+64t=1$ where, $t=x+11z$

$⟹7z+(7\times 9t+t)=1$

$⟹t+7(z+9t)=1$

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

Taking $u=0,t=1$

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

$t=x+11z$ gives $x=100$

$z=x-y$ gives $y=109$

all solutions are given by

$x=100+711k$

$y=109+775k,$ k is an integer

Putting $k=o,$ the least positive integral solution is $x=100,y=109$