Mysterious Light（数学）

问题 B: Mysterious Light

时间限制: 2 Sec   内存限制: 256 MB
提交: 208   解决: 54
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题目描述

Snuke is conducting an optical experiment using mirrors and his new invention, the rifle of Mysterious Light.

Three mirrors of length N are set so that they form an equilateral triangle. Let the vertices of the triangle be a,b and c.

Inside the triangle, the rifle is placed at the point p on segment ab such that ap=X. (The size of the rifle is negligible.) Now, the rifle is about to fire a ray of Mysterious Light in the direction of bc.

The ray of Mysterious Light will travel in a straight line, and will be reflected by mirrors, in the same ways as "ordinary" light. There is one major difference, though: it will be also reflected by its own trajectory as if it is a mirror! When the ray comes back to the rifle, the ray will be absorbed.

The following image shows the ray's trajectory where N=5 and X=2.

It can be shown that the ray eventually comes back to the rifle and is absorbed, regardless of the values of N and X. Find the total length of the ray's trajectory.

Constraints
2≦N≦1012
1≦X≦N−1
N and X are integers.
Partial Points
300 points will be awarded for passing the test set satisfying N≦1000.
Another 200 points will be awarded for passing the test set without additional constraints.

输入

The input is given from Standard Input in the following format:N X

输出

Print the total length of the ray's trajectory.

样例输入

5 2

样例输出

12

题意：从正三角形的一条边沿bc方向发出一道光线，光线遇见正三角形的边或者自己的轨迹都能发生反射，直到回到起点，求光路的长度。

题解：可以模拟一下，发现问题存在子问题，每次反射后构成的平行四边形的边长N能否取余放射出的正三角形的边长X(不同于一开始的n和x），若能取余，则最终一定可以从平行四边形的入口出去(就是一开始反射的起点），则可一步步递归到最终能取余的情况，再返回来。

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