What is the smallest possible natural number \(n\) for which the equation \(x^2 - nx + 2014 = 0\) has integer roots?

This note is part of the set Pre-RMO 2014

What is the smallest possible natural number \(n\) for which the equation \(x^2 - nx + 2014 = 0\) has integer roots?

This note is part of the set Pre-RMO 2014

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TopNewestLet the roots of the equation be \(\alpha\) and \(\beta\).

Then \(n = \alpha + \beta\) and \(2014 = \alpha \beta\).

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

Therefore, the possible values of \(n\) are \(2015, 1009, 125\) and \(91\). The minimum value is \(91\), so \(n = \boxed{91}\) – Pranshu Gaba · 2 years, 7 months ago

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Is the answer 91? – Abhineet Nayyar · 2 years, 7 months ago

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– Krishna Ar · 2 years, 7 months ago

Yes , it is 91 :DLog in to reply

– Abhineet Nayyar · 2 years, 7 months ago

Yayy!! lol:D:DLog in to reply

91 – Gaurav Singh · 1 year, 9 months ago

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91 – Akshat Sharda · 1 year, 9 months ago

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91 – Sahil Nare · 2 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 · 2 years, 7 months ago

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Explain? – Shiv Kumar · 2 years, 7 months ago

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