On a traffic signal, traffic light charges its colour after every 24, 30 and 36 seconds in green, red and orange light. How many times in an hour only green and red light will change simmultaneously.

Answer

To solve the problem of how many times the green and red lights change simultaneously in one hour, we can follow these steps:

### Step 1: Determine the time intervals for each light
- Green light changes every 24 seconds.
- Red light changes every 30 seconds.
- Orange light changes every 36 seconds.

### Step 2: Find the Least Common Multiple (LCM) of the intervals
To find out when the green and red lights change simultaneously, we need to calculate the LCM of the time intervals for the green and red lights (24 seconds and 30 seconds).

**Calculation of LCM:**
- The prime factorization of 24 is \(2^3 \times 3^1\).
- The prime factorization of 30 is \(2^1 \times 3^1 \times 5^1\).

To find the LCM, we take the highest power of each prime factor:
- For \(2\), the highest power is \(2^3\).
- For \(3\), the highest power is \(3^1\).
- For \(5\), the highest power is \(5^1\).

Thus, the LCM is:
\[ LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \]

### Step 3: Calculate how many times the lights change in one hour
One hour is equal to 3600 seconds. To find out how many times the green and red lights change simultaneously in one hour, we divide the total seconds in an hour by the LCM we just calculated:

\[ \text{Number of changes} = \frac{3600 \text{ seconds}}{120 \text{ seconds}} = 30 \]

### Step 4: Adjust for the orange light
Now, we need to consider that the orange light also changes every 36 seconds. We need to find the times when both the green and red lights change, excluding when the orange light changes.

To find the LCM of the intervals for the green and red lights with the orange light, we calculate the LCM of 24, 30, and 36.

**Calculation of LCM:**
- The prime factorization of 36 is \(2^2 \times 3^2\).

Now, we take the highest powers:
- For \(2\), the highest power is \(2^3\) (from 24).
- For \(3\), the highest power is \(3^2\) (from 36).
- For \(5\), the highest power is \(5^1\) (from 30).

Thus, the LCM is:
\[ LCM = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360 \]

### Step 5: Calculate how many times the green and red lights change excluding orange
Now, we find how many times the green, red, and orange lights change simultaneously in one hour:

\[ \text{Number of changes} = \frac{3600 \text{ seconds}}{360 \text{ seconds}} = 10 \]

### Step 6: Subtract the simultaneous changes from the total changes
Since the green and red lights change together 30 times, and they change together with the orange light 10 times, we subtract the simultaneous changes from the total changes:

\[ \text{Only green and red changes} = 30 - 10 = 20 \]

### Final Answer
Thus, the green and red lights will change simultaneously **20 times** in one hour.

---