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

#Algebra #Pre-RMO

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Comments

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 $n = \boxed{91}$

Pranshu Gaba - 6 years, 8 months ago

Is the answer 91?

A Former Brilliant Member - 6 years, 8 months ago

Yes , it is 91 :D

Krishna Ar - 6 years, 8 months ago

Yayy!! lol:D:D

A Former Brilliant Member - 6 years, 8 months ago

Explain?

Shiv Kumar - 6 years, 8 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.

Sanchit Ahuja - 6 years, 7 months ago

Sahil Nare - 6 years, 1 month ago

Akshat Sharda - 5 years, 10 months ago

gaurav singh - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$