Did you say the answer was 20 seconds - the LCM of 1, 2, 4 and 5? I think it is not 20 but 30 seconds. It is the one second for which the words glow, that is causing all the mischief.
Just for the ease of understanding, imagine that each word immediately upon lighting up lets off a small beep. The first word, therefore, gives a beep on lighting up. It remains glowing for one second and, then, goes off for one second. Then, it glows again after making a beep. So, the interval between the two beeps is two seconds.
Similarly, the second word gives a beep on lighting up; remains glowing for one second; and, then, goes off for two seconds. It lights up again, giving a beep. In this case, the interval between two beeps is three seconds.
Likewise, the interval between two beeps by the third word will be five seconds, and by the fourth word will be six seconds.
All the four words will light up simultaneously after beeping at the same time. That will happen after 30 seconds - the LCM of 2, 3, 5 and 6. So, I think the answer is 30 seconds. All the four words will glow together at intervals of 30 seconds.
You don't think I am right? Okay, I have tried to create a table. The time under the "Time" column is in seconds; I am not interested in the hour and minute part. Thus, initially, all the words are in OFF position. From 01 to 02 seconds, the first words glows, and the others are OFF; from 02 to 03 seconds, the first word goes OFF for a second and the others are also OFF. That is how it continues. This is what I get: