An Algebra Problem

Number Theory Level 3

p + 1 q + 1 r + 1 s = 89 68 \Large p + \cfrac{1}{q+\cfrac{1}{r+\cfrac{1}{s}}} = \frac{89}{68}

Given that p , q , r , p, q, r, and s s are natural numbers such that they satisfy the equation above, find the value of p q + r s pq + rs .


The answer is 23.

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Rohith M.Athreya by Rohith M.Athreya , 22, India

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2 solutions

Adarsh Kumar
Nov 5, 2015

89 68 = 1 + 21 68 = 1 + 1 68 21 = 1 + 1 3 + 5 21 = 1 + 1 3 + 1 21 5 = 1 + 1 3 + 1 4 + 1 5 \dfrac{89}{68}=1+\dfrac{21}{68} =1+\dfrac{1}{\dfrac{68}{21}} =1+\dfrac{1}{3+\dfrac{5}{21}} =1+\dfrac{1}{3+\dfrac{1}{\dfrac{21}{5}}} =1+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{5}}}

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Wow. Liked this apporoach

Akhil Krishna - 5 years, 7 months ago

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This is continued fraction in N. Theory...

Shubham Ghosh - 5 years, 6 months ago
Deepak Kumar
Nov 4, 2015

Hint:Try to break the RHS in same pattern as LHS.89/68=1+21/68.So p=1.Now use 21/68 wisely and proceed.

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