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# Pre-RMO 2014/6

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

Note by Pranshu Gaba
2 years, 3 months ago

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Let 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 $$n = \boxed{91}$$ · 2 years, 3 months ago

Is the answer 91? · 2 years, 3 months ago

Yes , it is 91 :D · 2 years, 3 months ago

Yayy!! lol:D:D · 2 years, 3 months ago

91 · 1 year, 5 months ago

91 · 1 year, 5 months ago

91 · 1 year, 9 months ago

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. · 2 years, 3 months ago