This is a brilliant puzzle! I was able to find a solution: the rules of the lights are like an arithmetic sequence, which any two adjacent numbers in the sequence have the same difference.
The lights are actually based on multiples of any number, or a common difference in the terms. For example, one of the lights light up when the input is 0, 5, 10, 15 etc., and perhaps another one goes 4, 13, 22, 31 etc. which will differ every time I press the ‘Reset’ button.
So far, in all the attempts I tried, I was unfortunately unable to find a number that lights up all four lights. For instance, the red and blue lights cannot light up at the same time because the red light only lights up when the number ends in a 4, and the blue light only lights up when it ends in 6. The number cannot end in 4 and 6 at the same time.
Overall, I would say that the light puzzle is really interesting because it makes me investigate different patterns for each specific light, and I really like finding patterns hidden in sequences.