Tours of the regional capital and the white house begin at 8.30 am from tour agency. Tours for the regional capital leave every 15
min. Tours for the white house leave every 20 min. After many minutes do the tours leave at the same time ?

Options

A. 60 min
B. 50 min
C. 1 hr 5 min
D. 15 min

Answer

To solve the problem of when the tours for the regional capital and the White House leave at the same time, we need to find the least common multiple (LCM) of the two time intervals: 15 minutes for the regional capital and 20 minutes for the White House.

### Step-by-Step Solution:

1. **Identify the time intervals**:
- Regional capital tour leaves every 15 minutes.
- White House tour leaves every 20 minutes.

2. **Find the prime factorization of each time interval**:
- For 15:
- 15 can be factored into prime numbers as \( 3 \times 5 \).
- For 20:
- 20 can be factored into prime numbers as \( 2^2 \times 5 \).

3. **List the prime factors**:
- The prime factors of 15 are \( 3^1 \) and \( 5^1 \).
- The prime factors of 20 are \( 2^2 \) and \( 5^1 \).

4. **Determine the LCM**:
- To find the LCM, take the highest power of each prime factor:
- For \( 2 \): the highest power is \( 2^2 \) (from 20).
- For \( 3 \): the highest power is \( 3^1 \) (from 15).
- For \( 5 \): the highest power is \( 5^1 \) (common in both).
- Therefore, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 \]

5. **Calculate the LCM**:
- Calculate \( 2^2 = 4 \).
- Multiply \( 4 \times 3 = 12 \).
- Finally, multiply \( 12 \times 5 = 60 \).

6. **Conclusion**:
- The tours will leave at the same time after 60 minutes.

### Final Answer:
The tours for the regional capital and the White House will leave at the same time after **60 minutes**.

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