26309번 - Etched Emerald Orbs 다국어

An archaeologist team found a tomb of the ancient tribe and discovered $2^{125}$ emerald orbs inside the tomb. The ancient tribe etched a numbers on each emerald orb. The archeologists spent two decades realizing that the ancient tribe etched each emerald orb with a unique number. Moreover, the numbers are from $1$ to $2^{125}$ in the ancient language.

Eddy, the only mathematician in the archaeologist team, recently figured out the relation between the number $k$ and the emerald orb numbered $k$. The weight of the emerald orb numbered $k$ is exactly $\frac{1}{k}$ grams. Since the number on each emerald orb is distinct from the number on any other emerald orb, there are no two emerald orbs having the same weight.

Eddy proposes a hypothesis: the ancient tribe used these emerald orbs to represent weight less than $1$ gram. It is trivial that the emerald orb numbered $k$ can represent $\frac{1}{k}$ gram. Then, Eddy tries to represent $\frac{2}{k}$ grams for $3 ≤ k ≤ 4 × 10^{18}$ with two emerald orbs. He successfully finds that the emerald orbs numbered $2$ and $6$ can represent $\frac{2}{3} = \frac{1}{2} + \frac{1}{6}$ grams. Similarly, the emerald orbs numbered $3$ and $15$ can represent $\frac{2}{5} = \frac{1}{3} + \frac{1}{15}$ grams.

Can you write a program to help Eddy to check whether two emerald orbs can represent $\frac{2}{k}$ grams for a given integer $k$? If there are multiple combinations of two emerald orbs representing $\frac{2}{k}$ grams, output the combination minimizing the sum of the numbers etched on them. If there is no such combination, output $-1$.

The input contains only one positive integer $k$.

If there is no solution, output $-1$. Otherwise, output two distinct integers $x$ and $y$ separated by a blank where $\frac{2}{k} = \frac{1}{x} + \frac{1}{y}$ and $1 ≤ x < y ≤ 2^{125}$. If there are multiple solutions, output the solution minimizing $x + y$.

• $3 ≤ k ≤ 4 × 10^{18}$.