Determine the number greater than 19 and less than 34 that is divisible by 6 and 10. Responses 60 60 - no response given 24 24 - no response given 20 20 - no response given 30

1 answer

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.

  1. 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 \(2^1 \times 3^1 \times 5^1 = 30\).
  2. 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:
      • \(19 < 30 < 34\)
  3. 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.