Linear diophantine equation
Solve the following diophantine equation: $539x+315y=91$
Developing the algorithm of Euclides we find that:
- $hcf(539,315)=7$ (which divides $91$, and therefore the equation has a solution).
- $s_5=-7$
- $t_5=12$
Therefore, the solutions to the equation are: $$x=\dfrac{c}{hcf(a,b)}s_5+\dfrac{b}{hcf(a,b)}k=\dfrac{91}{7}(-7)+\dfrac{315}{7}k=-91+45k$$ $$y=\dfrac{c}{hcf(a,b)}t_5+\dfrac{a}{hcf(a,b)}k=\dfrac{91}{7}(12)+\dfrac{539}{7}k=156+77k$$ for any integer $k$.
$x=-91+45k; \ \ $ $y=156+77k$