Question

на маяке установлены три лампы. Первая вспыхивает один раз в 6 секунд, вторая один раз в 14 секунд, третья один раз в 21 секунд. В какой то момент все три лампы вспыхнули одновременно. Через сколько секунд после этого все три лампы вспыхнут одновременно в следующий раз?Решение пожалуйста

Answer 1

Problem Analysis

We are given three lamps on a lighthouse that flash at different intervals. We need to determine how long it will take for all three lamps to flash simultaneously again after they initially flash together.

Solution

To solve this problem, we need to find the least common multiple (LCM) of the three intervals at which the lamps flash. The LCM will give us the time it takes for all three lamps to flash together again.

Let's calculate the LCM of the three intervals: 6 seconds, 14 seconds, and 21 seconds.

Calculating the LCM

To find the LCM, we can use the prime factorization method or the division method. Let's use the division method for simplicity.

We start by dividing each interval by their common factors until we can no longer divide them evenly.

For 6 seconds: - Divide by 2: 6 ÷ 2 = 3

For 14 seconds: - Divide by 2: 14 ÷ 2 = 7

For 21 seconds: - Divide by 3: 21 ÷ 3 = 7

Now, we multiply all the divisors and the remaining numbers together to find the LCM:

2 × 3 × 7 × 7 = 294

Therefore, the LCM of 6 seconds, 14 seconds, and 21 seconds is 294 seconds.

Answer

After all three lamps initially flash together, it will take 294 seconds for them to flash together again.

Verification

Let's verify this result by checking if all three lamps will flash together after 294 seconds.

- The first lamp flashes every 6 seconds. After 294 seconds, it will have flashed 294 ÷ 6 = 49 times. - The second lamp flashes every 14 seconds. After 294 seconds, it will have flashed 294 ÷ 14 = 21 times. - The third lamp flashes every 21 seconds. After 294 seconds, it will have flashed 294 ÷ 21 = 14 times.

Since all three lamps have flashed the same number of times, they will flash together again after 294 seconds.

Conclusion

After all three lamps on the lighthouse initially flash together, they will flash together again after 294 seconds.