Planets X, Y and Z take 360, 450 and 540 days, respectively, to rotate around the same sun. If the three planets are lined up in a ray having the sun as its endpoint, what is the minimum positive number of days before they are on the same line again?

a little hint i got recently is that the planets dont needa be on the original starting line, they could line up at the opposite side of the sun as long as they're in a line

 Apr 5, 2024

Yes, in some point, the planets will line up in a line that is not in the original starting line.


The key to this problem is finding the least common multiple (LCM) of 360, 450, and 540. This represents the smallest number of days that will take for all three planets to complete one full rotation around the sun relative to their starting positions.


Here's why:


If all planets complete their rotations in multiples of their individual periods (e.g., 360 days for X, 900 days for Y, etc.), they might not be lined up



The LCM ensures that after that specific number of days, X will have completed exactly m rotations (for some integer m), Y will have completed exactly n rotations (for some integer n), and Z will have completed exactly o rotations (for some integer o).


To find the LCM, we can use various methods. In this case, since the numbers are relatively close, you might be able to find it by inspection.


However, a more general approach is to use the Euclidean Algorithm:


Find the greatest common divisor (GCD) of the largest two numbers (here, 450 and 540).


Divide the LCM of the initial three numbers (which doesn't exist yet) by the GCD you just found. This gives you the LCM of the first two numbers.


Now, find the GCD of the LCM you obtained in step 2 and the remaining number (here, 360).


The final LCM is the product of the GCDs you found in steps 1 and 3.


Following these steps, you'll find the GCD of 450 and 540 to be 90. The LCM of 450 and 540 is then (450 * 540) / 90 = 300.


Finally, the GCD of 300 and 360 is 60. Therefore, the LCM of 360, 450, and 540 is (300 * 60) = 18000.


However, 18000 is not the minimum positive number of days. Why? Because all three planets might be lined up again at a smaller common multiple. We need to find the least common multiple.


Looking closer, we see that 540 (Z's rotation period) is divisible by both 360 (X's period) and 450 (Y's period).


This means that every time Z completes a full rotation, X will have completed some integer number of rotations and Y will have completed some other integer number of rotations. In other words, they will all be lined up again after every 540 days.


Therefore, the minimum positive number of days before they are on the same line again is 540​ days.

 Apr 9, 2024

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