There are three lights Red, Green and Orange. The lights will be ON after intervals of 18 seconds, 22 seconds and 34 seconds respectively and each light is put off after 2 seconds. If all the lights were ON together now, and all lights were simultaneously got OFF after 't' seconds, where 600 < t < 800, find 't'?

To solve this problem, we need to find the least common multiple (LCM) of the intervals at which the lights turn on, then determine a value of \( t \) that falls within the range \( 600 < t < 800 \).

### Step 1: Identify the intervals at which the lights turn on
- Red light: 18 seconds
- Green light: 22 seconds
- Orange light: 34 seconds

### Step 2: Calculate the LCM of these intervals
The LCM of 18, 22, and 34 is the smallest number that is a multiple of all three.

- **Prime factorization**:
  - \( 18 = 2 \times 3^2 \)
  - \( 22 = 2 \times 11 \)
  - \( 34 = 2 \times 17 \)

- **LCM calculation**:
  - \( \text{LCM} = 2 \times 3^2 \times 11 \times 17 = 2 \times 9 \times 11 \times 17 = 3366 \)

Thus, the LCM of 18, 22, and 34 is 3366 seconds.

### Step 3: Find the multiple of the LCM within the given range
We are given that \( 600 < t < 800 \). To find \( t \), we need to check the smallest multiple of 3366 that falls within this range.

\[
\text{Multiple of } 3366 \text{ closest to this range is } \frac{600}{3366} \text{ and } \frac{800}{3366}.
\]

Since the multiple of the LCM of the intervals is greater than the upper limit of 800 seconds, let's check for the first occurrence within the given range:
\[
k = 1: \text{LCM} \times k = 3366 
\]

Since \( k = 1 \) is already much larger, we need to divide by the smallest factor which brings us into the desired range, 3366, still requires a smaller multiple in the relevant range.

### Final Answer:
No multiple falls between 600 and 800 seconds, thus any multiple or common set must be a zero or matching factor into the interval without leaving remainder.

Do you want to dive deeper into calculation steps? 

1. Why are prime factors relevant?
2. How LCM works in periodic problems?
3. What alternative methods exist for periodic matching?
4. How might this apply to another range?
5. Can it extend to other periodic sequences?

**Tip**: Consider reviewing modular arithmetic; it is very useful in periodic problems.

Learn how to solve a problem involving periodic lights turning on and off using concepts like Least Common Multiple (LCM) and modular arithmetic. Discover the application of LCM to synchronize multiple intervals and find a suitable 't' within a specified range. Suitable for advanced high school students and those interested in number theory and modular arithmetic.

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