Determine the smallest 3 digit number which is exactly divisible by 6 8 and 12

General
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determine the smallest 3 digit number which is exactly divisible by 6 8 and 12

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Determine the smallest 3 digit number which is exactly divisible by 6, 8, and 12

Answer: To find the smallest three-digit number which is exactly divisible by 6, 8, and 12, we can follow these steps:

  1. Find the Least Common Multiple (LCM) of 6, 8, and 12:

    • First, find the prime factorizations of each number:

      6 = 2 \times 3
      8 = 2^3
      12 = 2^2 \times 3

    • Determine the LCM by taking the highest power of each prime that appears in these factorizations:

      \text{LCM} = 2^3 \times 3 = 8 \times 3 = 24

  2. Identify the smallest three-digit number divisible by 24:

    • The smallest 3-digit number is 100.
    • Divide 100 by 24 to determine the quotient and whether it is a whole number:
      100 \div 24 \approx 4.167
    • Since 4.167 is not a whole number, multiply the next whole number greater than 4.167 by 24:
      • The smallest integer greater than 4.167 is 5.
      24 \times 5 = 120

  3. Verification:

    • Check that 120 is indeed the smallest 3-digit number divisible by 24 by verifying divisibility:
      • Since 120 is greater than 100 and is a multiple of 24, it fits the criteria.

Final Answer: The smallest 3-digit number which is exactly divisible by 6, 8, and 12 is \boxed{120}.