## 6. Light Bulbs

In the room, there are two light bulbs. When the switch of the first light bulb is turned on, it lights up after 6 seconds and stays on for 5 seconds, then it is off for 6 seconds and on for 5 seconds, and this cycle repeats continuously. When the switch of the second light bulb is turned on, it lights up after 4 seconds and stays on for 3 seconds, then it is off for 4 seconds and on for 3 seconds, and this cycle repeats continuously. Linda turned on both switches at the same time and turned them off after 2021 seconds. How many seconds did both light bulbs shine simultaneously during this time?

## Result: 392
## Solution.

Every 11 seconds, the first light will repeat a cycle where it will not light for 6 seconds and then light for 5 seconds. The second light will repeat such a cycle every 7 seconds.

Since $V(11,7)=77$, the time interval we need to observe lasts 77 seconds.

In the first two rectangles, yellow represents the seconds during which the lights are on, and gray represents the seconds during which the lights are off. The first rectangle shows the state of the first light, and the second shows the state of the second light. In the third rectangle, the seconds when both lights are on at the same time are marked.

In the observed period of 77 seconds, both lights are on at the same time for $1+3+1+3+2+2+3=$ 15 seconds. Since $2021=26 \cdot 77+19$, there will be 26 such periods, and then there will be 19 seconds remaining. In these 19 seconds, the lights will be on at the same time for 2 seconds.

Both lights are on at the same time for a total of $26 \cdot 15+2=392$ seconds.