163233

I started by making a table with rows indicating the numbers starting from 1 and columns indicating Red, Yellow, Green and Blue colours. I starting putting the numbers from 1 to 20 one after another into the box and marked the results on my table by putting 'x'.
By the time I reached 10, it was clear that Green glowed for all even numbers.
By the time I reached 16, it was clear that Red glowed when I typed a number that was multiple of 5 plus 1, example, 15+1=16.
Yellow was tricky, but by the time I reached 28, I saw that the gap between the numbers for which yellow glowed increased by 1 starting with number 1.
Blue was the trickiest, I could only figure out the pattern by 25 that the gap between the numbers for which blue glowed increased by 2 starting from number 1
Then, by the above patterns, 36 was the first number which made all lights glow