A number theory problem by Kshetrapal Dashottar

Number Theory Level 3

Find the number of fractions which can be written simultaneously in the forms 7 k − 5 5 k − 3 \dfrac{7k-5}{5k-3} and 6 l − 1 4 l − 3 \dfrac{6l-1}{4l-3} for some integers k k and l l .


The answer is 8.

Kshetrapal Dashottar by Kshetrapal Dashottar , 20, India

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1 solution

5k – 3

7k – 5

4 – 3 6 – 1   28k – 21k – 20 + 15 = 30k – 5k – 18 + 3 2k + 16k + 2 – 12 = 0 k + 8k +  – 6 = 0 k + 8k +  = 6 k( + 8) +  + 8 = 14 (k + 1)( + 8) = 14 (k + 1)( + 8) = 14 = 14 × 1 = 7 × 2 = – 14  – 1 = –7  – 2 if k + 1 = 14 and + 8 = 1 or k + 1 = 1 and  + 8 = 14 (k = 13,  = – 7) (k = 0 ,  = 6) in the same way we can find the other solution (k, ) = (13, – 7), (– 15, – 9), (0, 6), (– 2, – 22), (6, – 6),(–8, – 10),(1, –1),(–3, –15) so total 8 solutions A

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