Solve the following diophantine equation: $$539x+315y=91$$
See development and solution
Development:
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 $$k$$.
Solution:
$$x=-91+45k; \ \ $$ $$y=156+77k$$