To find a number that is greater than 19 and less than 34 that is divisible by both 6 and 10, we first need to determine the least common multiple (LCM) of the two numbers.
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Finding the LCM of 6 and 10:
- The prime factorization of 6 is \(2 \times 3\).
- The prime factorization of 10 is \(2 \times 5\).
- The LCM is found by taking the highest power of each prime factor:
- For 2, the highest power is \(2^1\).
- For 3, the highest power is \(3^1\).
- For 5, the highest power is \(5^1\).
- Thus, the LCM is \(2^1 \times 3^1 \times 5^1 = 30\).
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Finding a number greater than 19 and less than 34:
- The LCM we calculated is 30. Now we check if it falls within the range:
- \(19 < 30 < 34\)
- The LCM we calculated is 30. Now we check if it falls within the range:
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Conclusion:
- 30 is the only number that meets all the criteria: it is greater than 19, less than 34, and divisible by both 6 and 10.
Therefore, the number you are looking for is 30.