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
6 years, 9 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 $n = \boxed{91}$

- 6 years, 9 months ago

Is the answer 91?

- 6 years, 9 months ago

Yes , it is 91 :D

- 6 years, 9 months ago

Yayy!! lol:D:D

- 6 years, 9 months ago

Explain?

- 6 years, 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.

- 6 years, 9 months ago

91

- 6 years, 3 months ago

91

- 5 years, 11 months ago

91

- 5 years, 11 months ago