For a natural number \(b\), let \(N(b)\) denote the number of natural numbers \(a\) for which the equation \(x^2 + ax + b = 0\) has integer roots. What is the smallest value of \(b\) for which \(N(b) =6\)?

This note is part of the set Pre-RMO 2014

For a natural number \(b\), let \(N(b)\) denote the number of natural numbers \(a\) for which the equation \(x^2 + ax + b = 0\) has integer roots. What is the smallest value of \(b\) for which \(N(b) =6\)?

This note is part of the set Pre-RMO 2014

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TopNewestThe question becomes in finding the smallest positive integer which has 12 factors(negative). – Lawrence Bush · 2 years, 7 months ago

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– Pranshu Gaba · 2 years, 7 months ago

We must also check the smallest integer with 11 factors, for example \(2^{10}\), even that will give \(N(b) = 6\).Log in to reply

Answer is 60 – Ar Agarwal · 2 years, 7 months ago

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