# Problems from Linear diophantine equation

Solve the following diophantine equation: $$539x+315y=91$$

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### 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$$

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