In order for all the lights to light up, the smallest number was 36. I didn't know the rule for when the different color lights light up so, I tested a few numbers and wrote which numbers lit up. After collecting data, I found a rule for each color:
Blue light: When I put each of the numbers 0,1,4,9, and 16 (ect) in the machine, the blue light shone. I found that the difference between the previous number and the following number, was the next larger odd number. For example, the difference of 0 to 1, is one. 1 to 4, is three. 4 to 9, is five. The difference changes and is the next odd number. The answers are also all square numbers.
Green light: I noticed that the green lights shine after every two numbers. In other words, you can find the next green light by adding 2 to a green light.
Yellow light: The difference between the two numbers change by increasing 1 each number in the pattern after. This pattern started at 1 and went on to 3,6,10 and so on. From 1 to 3, you are adding 2. From 3 to 6, you are adding 3. From 6 to 10, you are adding 4. The difference between two numbers change by increasing by 1.
Red light: The red lights occur every 5 numbers. This pattern starts at 1 so, all you have to do is add 5 to the next red light destination.
Finally, I wrote down all the numbers until the twentieth number in each of the patterns. I chose to write 20 numbers because the common number for each pattern might not appear in the beginning. I followed each rule one by one until I got four long lists of 20 numbers each. I compared each line to try to find one number that all the lists had. In order to find the smallest, I looked closest to the beginning of the lists where the numbers were the smallest. At the end, the smallest common number I found was 36.