A certain robot only moves forward or turns right, at an angle of $x$ degrees relative to the original direction it was moving in, as shown in the figure below:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-09.jpg?height=254&width=811&top_left_y=564&top_left_x=725)

To return to the original direction and sense, the robot needs to turn right a certain number of times. For example, if $x=90^{\circ}$, then the robot needs to turn right four times:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-09.jpg?height=399&width=829&top_left_y=1057&top_left_x=722)

a) How many times does the robot need to turn right if $x=60^{\circ}$?

b) How many times does the robot need to turn right if $x=42^{\circ}$?

c) And if $x=47^{\circ}$?
Solution

a) Each time the robot turns right, the direction in which it moves varies by $60^{\circ}$ clockwise. Thus, the total variation in the direction of its movement after a certain number of turns is always a multiple of $60^{\circ}$.

After six turns, the total variation in the direction of its movement will be $6 \times 60^{\circ}=360^{\circ}$ clockwise. However, rotating by an angle of $360^{\circ}$ means returning to the original direction and sense. Therefore, after six turns, the direction and sense of the robot's movement will be the same as the original direction and sense.

b) The total variation in the direction of the robot's movement after a certain number of turns is always a multiple of $42^{\circ}$. Note that $360^{\circ}$ is not a multiple of $42^{\circ}$. Thus, it is impossible for the robot to vary the direction and sense of its movement by $360^{\circ}$ by turning right each time by an angle of $42^{\circ}$.

However, as soon as the total variation in the angle of its movement is equal to a multiple of $360^{\circ}$, the robot will have returned to the original direction and sense of movement. Therefore, we need to find the smallest number that is a multiple of both $42^{\circ}$ and $360^{\circ}$.

Noting that $42=6 \times 7$ and that $360=6^{2} \times 10$, we have that the least common multiple (LCM) between 42 and 360 is equal to $6^{2} \times 7 \times 10=2520$.

Since $2520^{\circ}=42^{\circ} \times 60$, it is necessary for the robot to turn right 60 times for the total variation in the direction of its movement clockwise to be equal to $2520^{\circ}$. Thus, only after 60 turns will the robot return to the original direction and sense of its movement.

c) As in the previous item, we need to find the smallest multiple of $47^{\circ}$ that is also a multiple of $360^{\circ}$. Since 47 and 360 are coprime, the LCM between them is equal to $47 \times 360$. This means that the robot must turn 360 times for its movement to return to the original direction and sense.

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