Three sprinters running on a circular track completes a lap in 4 minutes,5 minutes and 7 minutes. Find the time after which they will meet for the 1st time at the starting point when all the three sprinters start running simultaneously.
Question
Three sprinters running on a circular track completes a lap in 4 minutes,5 minutes and 7 minutes. Find the time after which they will meet for the 1st time at the starting point when all the three sprinters start running simultaneously.
Solution
To solve this problem, we need to find the least common multiple (LCM) of the three given times (4 minutes, 5 minutes, and 7 minutes). The LCM of a set of numbers is the smallest positive integer that is divisible by each of the numbers in the set.
Here are the steps to find the LCM:
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First, list the prime factors of each number:
- 4 = 2 * 2
- 5 = 5
- 7 = 7
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Then, for each prime number, take the highest power that appears in any of the factorizations. In this case, we have:
- For 2, the highest power is 2 (from 4 = 2 * 2)
- For 5, the highest power is 1 (from 5 = 5)
- For 7, the highest power is 1 (from 7 = 7)
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Multiply these highest powers together:
- 2^2 * 5^1 * 7^1 = 4 * 5 * 7 = 140
So, the LCM of 4, 5, and 7 is 140. This means that the three sprinters will meet for the first time at the starting point 140 minutes after they start running.
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