Pre-RMO 2014/6

What is the smallest possible natural number nn for which the equation x2nx+2014=0x^2 - nx + 2014 = 0 has integer roots?


This note is part of the set Pre-RMO 2014

Note by Pranshu Gaba
5 years ago

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Let the roots of the equation be α\alpha and β\beta.

Then n=α+βn = \alpha + \beta and 2014=αβ2014 = \alpha \beta.

The possible integral pairs of (α,β)(\alpha, \beta) are (1,2014),(2,1007),(19,106),(1, 2014), (2, 1007), (19, 106), and (38,53)(38, 53).

Therefore, the possible values of nn are 2015,1009,1252015, 1009, 125 and 9191. The minimum value is 9191, so n=91n = \boxed{91}

Pranshu Gaba - 5 years ago

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Is the answer 91?

Abhineet Nayyar - 5 years ago

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Yes , it is 91 :D

Krishna Ar - 5 years ago

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Yayy!! lol:D:D

Abhineet Nayyar - 5 years ago

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Explain?

Shiv Kumar - 5 years ago

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The discriminant of the above equation should be a perfect square for the equation to have integral roots. So the smallest value of n for which it is a perfect square is 91.

Sanchit Ahuja - 5 years ago

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91

Sahil Nare - 4 years, 5 months ago

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91

Akshat Sharda - 4 years, 2 months ago

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91

gaurav singh - 4 years, 2 months ago

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