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


Solution: If a fraction is simultaneously in the forms $\frac{7 k-5}{5 k-3}$ and $\frac{6 l-1}{4 l-3}$, we must have

$$
\frac{7 k-5}{5 k-3}=\frac{6 l-1}{4 l-3}
$$

This simplifies to $k l+8 k+l-6=0$. We can write this in the form

$$
(k+1)(l+8)=14
$$

Now 14 can be factored in 8 ways: $1 \times 14,2 \times 7,7 \times 2,14 \times 1,(-1) \times(-14),(-2) \times(-7)$, $(-7) \times(-2)$ and $(-14) \times(-1)$. Thus we get 8 pairs:

$$
(k, l)=(13,-7),(6,-6),(1,-1),(0,6),(-15,-9),(-8,-10),(-3,-15),(-2,-22)
$$

These lead respectively to 8 fractions:

$$
\frac{43}{31}, \quad \frac{31}{27}, \quad 1, \quad \frac{55}{39}, \quad \frac{5}{3}, \quad \frac{61}{43}, \quad \frac{19}{13}, \quad \frac{13}{9}
$$
