<s>[INST] All fractions which can be written simultaneously in the forms $\frac{7k-5}{5k-3}$ and $\frac{6l-1}{4l-3}$ Find all fractions which can be written simultaneously in the forms $\frac{7k-5}{5k-3}$ and $\frac{6l-1}{4l-3}$
for some integers $k,l$.

Please check my answer and tell me is correct or not....
$$\frac{43}{31},\frac{31}{27},1,\frac{55}{39},\frac{5}{3},\frac{61}{43},\frac{19}{13},\frac{13}{9}$$

 [/INST] We have that $$\frac{7k-5}{5k-3}=\frac{6l-1}{4l-3}\iff kl+8k+l=6.$$ That is, if $k\ne -1,$
$$l=2\frac{3-4k}{k+1}=-2\left(4-\frac{7}{k+1}\right)=-8+\frac{14}{k+1}.$$ Since $l$ has to be an integer $k+1$ must divide $14.$ So, we have that $k\in\{-15,-8,-3,-2,0,1,6,13\}.$ 
Note that $k\ne -1$ since if $k=-1$ the equation $kl+8k+l=6$ gives $-8=6$ which doesn't hold.
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