A number theory problem by Vishnu Kadiri

Number Theory Level 4

How many fractions are there such that they can be written simultaneously in the forms

7 k − 5 5 k − 3 and 6 l − 1 4 l − 3 \dfrac { 7k-5 }{ 5k-3 } \quad \text{and} \quad \dfrac { 6l-1 }{ 4l-3 }

for some integers k k and l ? l?


The answer is 8.

Vishnu Kadiri by Vishnu Kadiri , 19, India

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1 solution

Rajen Kapur
Oct 8, 2016

Equate and cross multiply to simplify the two expressions, getting: k l + 8 k + l = 6 → ( k + 1 ) ( l + 8 ) = 14 kl+8k+l=6\rightarrow(k+1)(l+8)=14 . Now 14 has factors 1, 2, 7, 14 and each can be + / -, so there are 8 solutions in all.

2 Helpful 1 Interesting 0 Brilliant 0 Confused

You need to show that the solutions are distinct.

Jesse Nieminen - 4 years, 8 months ago

Well I have a genuine doubt...Please. reply soon.....We have kl + 8k + (l-6)=0 Since k,l are real, Discriminant>0 So 64≤4(l-6) implying (l-8)(l+2)≤0 So l belongs to the interval [8,-2] By plugging values of l and finding the corresponding values of k we get only three integral solution.. So the only three fraction are 1,5/3,61/43.... Tell me why didn't I get the rest solutions...Reply...Help..

Anubhav Mahapatra - 3 years, 8 months ago

@Jesse Nieminen Reply to your comment: It is unnecessary to verify distinctness as it is a linear two-dimensional transformation which is one-to-one.

Rajen Kapur - 4 years, 8 months ago

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You should have said that in your solution.

Jesse Nieminen - 3 years, 8 months ago

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