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In a high -rise building with 84 floors,...

In a high -rise building with 84 floors, one delivery boy delivers at every `6^(th)` floor and the other at every `8^(th)` floor. If the two start delivering together from the entrance of the building, at which floors will they meet again?

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To solve the problem, we need to find the floors where both delivery boys will meet again after starting together from the entrance of the building. One delivery boy delivers at every 6th floor, and the other at every 8th floor. ### Step-by-Step Solution: 1. **Identify the delivery patterns**: - The first delivery boy delivers at every 6th floor: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84. - The second delivery boy delivers at every 8th floor: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88 (but we only consider up to 84). 2. **Find the Least Common Multiple (LCM)**: - To find the floors where both boys meet, we can calculate the LCM of 6 and 8. - The prime factorization of 6 is \(2^1 \times 3^1\). - The prime factorization of 8 is \(2^3\). - The LCM is found by taking the highest power of each prime factor: - For 2: \(2^3\) (from 8) - For 3: \(3^1\) (from 6) - Therefore, LCM = \(2^3 \times 3^1 = 8 \times 3 = 24\). 3. **List the multiples of the LCM within the range of floors**: - We now find the multiples of 24 that are less than or equal to 84: - \(24 \times 1 = 24\) - \(24 \times 2 = 48\) - \(24 \times 3 = 72\) - \(24 \times 4 = 96\) (not included since it exceeds 84) 4. **Conclusion**: - The floors where both delivery boys will meet again are: **24, 48, and 72**. ### Final Answer: The delivery boys will meet again at the floors: **24, 48, and 72**.
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