Find all fractions which can be written simultaneously reduced in the forms (7K - 5)/(5K - 3) and (6L - 1)/(4L - 3) for some integers K, L.
I confess to being confused by the term "simultaneously reduced".
If the fractions need to be identical before simplifying, then (7K-5)=(6L-1) and (5K-3)=(4L-3).
The only solution to these simultaneous equations is K = -8, L = -10
This makes the fractions both (-61)/(-43).
Is that "simultaneously reduced"? I don't think so.
Or maybe the two fractions just need to equal after being simplified?
Then, cross multiplying, (7K-5)*(4L-3) = (6L-1)*(5K-3).
Solving for L gives L = 14/(K+1) - 8
K+1 can only be 14,7,2,1,-1,-2,-7, or -14.
So the solutions are
K L Reduced Fraction
=== == ================
13 -7 43/31
6 -6 37/27
1 -1 1
0 6 5/3
-2 -22 19/13
-3 -15 13/9
-8 -10 61/43
-15 -9 55/39
Simultaneously reduced? (spoiler)