三行星再次与太阳共线

Planets Aligned

专题: Brainteaser / 脑筋急转弯
难度: L2
来源: QuantQuestion

题目详情

某恒星系中有三颗行星绕太阳公转，假设它们同向、同一平面。

三颗行星的公转周期分别为 40 年、60 年、90 年。今天它们与太阳共线。

问：下一次三颗行星与太阳再次共线（允许在太阳同侧或异侧）是在多少年后？

There is a solar system with three planets orbiting around the sun, assume they all orbit in the same direction and on the same plane. The first planet has a translation period of $40$ years, the second has one of $60$ years and another one of $90$ years. Today, the three planets are aligned with the sun. When is the next time, in years, the three planets will be aligned with the sun?

解析

若允许在太阳两侧也算共线，则“共线”的相对周期为半周期：40、60、90 分别对应 20、30、45。

求最小公倍数：

\mathrm{lcm}(20,30,45)=180.

因此下一次共线在 180 年后。

Original Explanation

The planets can be aligned either on the same or opposite side of the sun in order to be aligned. Therefore, the periods are really $20, 30,$ and $45$ . We want the least common multiple of these, which is simply $180$ .