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 $k, l$.
Problem
Source: CRMO 2015 region 1 p3
Tags: number theory, Integer, Fractions
khan.academy
30.09.2018 05:28
Are both reduced ones? If so, then just solving two simultaneous equation gives, $$k=-8, l=-10 \implies \textsc{fraction}=\frac{61}{43}$$
parmenides51
30.09.2018 11:27
yes, they are both reduced, there are 7 more solutions
NikoIsLife
30.09.2018 12:39
$$\frac{7k- 5}{5k - 3}=\frac{6l - 1}{4l - 3}\implies k l + 8 k + l - 6=0\implies(k+1)(l+8)=14$$This gives $(k,l)=(0,6),(1,-1),(6,-6),(13,-7),(-2,-22),(-3,-15),(-8,-10),(-15,-9).$
Substituting each of these to the equation gives the desired fractions.
Mathsblogger
14.04.2019 14:29
Nice solution
Aritra_dey
07.09.2019 17:59
nice one
Commander_Anta78
27.01.2021 21:25
i dont understand what khan.academy mean
TuZo
27.01.2021 21:54
Commander_Anta78 wrote: i dont understand what khan.academy mean This is only a partial solution. For the full solution see #4
sanyalarnab
12.07.2021 19:38
In the exam do they(participants) needed to write down all the values of those fractions because k,l can be found by solving as NikolsLife did but calculating all the fractions one by one is just horrible.
stretfordend
04.05.2023 20:41
there are 8 solutions we get (7k-5)(4l-3)=(6l-1)(5k-3) on simplifying , we get kl + l + 8k = 6 on factorizing we get (8+l)(k+1)=14 try all possible values