Answer: 24
If red 5 years, answer 120 years!

 

How to get the solution:
The red planet is back in its place every 1,2,3,4,5,6,7... Years
Blue is every 6,12,18,24... Years
Green is every 8,16,24... Years
Need to find the lowest number that appears in each list, which is 24.

 

If red 5 years, need lowest number on the following lists:
R: 5,10,15,20,...
B: 6,12,18,24,...
G: 8,16,24,...
Could write them all out but that's a lot of numbers! Instead think about what the numbers in the first list have in common - they all end in a 0 or a 5, so our answer must do too. All numbers in the B and G lists are even, so our number must be even so end in a 0. Now we can just list the multiples of 6 and 8 which end in 0:
B: 30, 60, 90, 120,...
G: 40, 80, 120,...
And 120 is the lowest.