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Question Number 19799 by Tinkutara last updated on 15/Aug/17

$$\mathrm{For}\mathrm{a}\mathrm{natural}\mathrm{number}b,\mathrm{let}N\left(b\right)\mathrm{denote}$$$$\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{natural}\mathrm{numbers}a\mathrm{for}$$$$\mathrm{which}\mathrm{the}\mathrm{equation}{x}^2+ax+b=0\mathrm{has}$$$$\mathrm{integer}\mathrm{roots}.\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{smallest}$$$$\mathrm{value}\mathrm{of}b\mathrm{for}\mathrm{which}N\left(b\right)=20?$$

Commented by dioph last updated on 16/Aug/17

$$x=\frac{-a\pm \sqrt{a^2-4b}}{2}$$$$x_1+x_2=-a$$$$x_1{x}_2=b$$$$N\left(b\right)=\frac{n}{2},\mathrm{where}n\mathrm{is}\mathrm{the}\mathrm{number}\mathrm{of}$$$$\mathrm{divisors}\mathrm{of}b$$$$n=40\Rightarrow b=p_1^{n_1}...p_k^{n_k}$$$$(n_1+1)...(n_k+1)=40$$$$\mathrm{The}\mathrm{smallest}\mathrm{number}\mathrm{with}40\mathrm{natural}$$$$\mathrm{divisors}\mathrm{I}\mathrm{can}\mathrm{find}\mathrm{is}$$$$b=2^4.3.5.7=1680$$

Commented by Tinkutara last updated on 16/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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