Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.

Let numbers be ' $\displaystyle x$ ' and ' $\displaystyle y$ ' such that $\displaystyle x y=289 \Rightarrow y=\frac{289}{x}$, ' $\displaystyle S$ ' be their sum, then \[S=x+y=x+\frac{289}{x} \] \[\frac{d S}{d x}=1-\frac{289}{x^{2}}, \frac{d S}{d x}=0 \Rightarrow x=17, \text { a positive integer } \] \[\left.\left.\frac{d^{2} S}{d x^{2}}\right]_{x=17}=289\left(\frac{2}{x^{3}}\right)\right]_{x=17}>0, \therefore S \text { is minimum when } x=17, y=17 \]