Problems from Linear diophantine equation

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: $$x={\displaystyle \frac{c}{hcf(a,b)}}{s}_5+{\displaystyle \frac{b}{hcf(a,b)}}k={\displaystyle \frac{91}{7}}(-7)+{\displaystyle \frac{315}{7}}k=-91+45k$$ $$y={\displaystyle \frac{c}{hcf(a,b)}}{t}_5+{\displaystyle \frac{a}{hcf(a,b)}}k={\displaystyle \frac{91}{7}}(12)+{\displaystyle \frac{539}{7}}k=156+77k$$ for any integer $k$.

Solution:

$x=-91+45k;$ $y=156+77k$

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