7. Lighthouse Alpha and Lighthouse Bravo flash their lights every 24 seconds and 30 seconds respectively. Given that they flash together at 9.15 p.m., how long will it take the lighthouses to flash their lights together again? 8. At a party, every 4th guest receives a key chain and every 9 th guest receives a mug. What is the position of the first guest who will receive both a key cha and a mug? At a party, every 3rd guest receives a key chain and every 8 th guest recei a mug. What is the position of the second guest who will receive both a k chain and a mug?

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UpStudy AI · Accepted Answer

**7. Lighthouse Flashing Together**

1. The two lighthouses flash every $$24$$ seconds and $$30$$ seconds respectively.
2. They will flash together every $$ \text{LCM}(24, 30) $$ seconds.
3. Factorize the numbers:
   - $$24 = 2^3 \times 3$$
   - $$30 = 2 \times 3 \times 5$$
4. The LCM is obtained by taking the highest power of each prime:
   $$
   \text{LCM}(24,30) = 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120 \text{ seconds}
   $$
5. Converting $$120$$ seconds to minutes:
   $$
   120 \text{ seconds} = 2 \text{ minutes}
   $$
6. Thus, the lighthouses will flash together again in $$2$$ minutes.

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**8. Party Guest Problems**

*Problem A:*

1. Every $$4$$th guest receives a key chain and every $$9$$th guest receives a mug.
2. The first guest to receive both is at position:
   $$
   \text{LCM}(4,9)
   $$
3. Factorize the numbers:
   - $$4 = 2^2$$
   - $$9 = 3^2$$
4. The LCM is:
   $$
   \text{LCM}(4,9) = 2^2 \times 3^2 = 4 \times 9 = 36
   $$
5. Therefore, the first guest to receive both a key chain and a mug is guest $$36$$.

*Problem B:*

1. Every $$3$$rd guest receives a key chain and every $$8$$th guest receives a mug.
2. The guests who receive both will be at positions that are multiples of:
   $$
   \text{LCM}(3,8)
   $$
3. Factorize the numbers:
   - $$3$$ is prime.
   - $$8 = 2^3$$
4. The LCM is:
   $$
   \text{LCM}(3,8) = 3 \times 2^3 = 3 \times 8 = 24
   $$
5. Thus, the guests receiving both items appear in the sequence:
   $$
   24, \, 48, \, 72, \dots
   $$
6. The first guest is at position $$24$$ and the second guest is at position:
   $$
   24 \times 2 = 48
   $$

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**Final Answers:**

7. $$2$$ minutes.  
8a. $$36$$th guest.  
8b. $$48$$th guest.
**7. Lighthouses Flash Together After 2 Minutes.** **8.** - **8a.** The first guest to receive both a key chain and a mug is the 36th guest. - **8b.** The second guest to receive both a key chain and a mug is the 48th guest.

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