Find the largest three-digit number that is completely divisible by 8, 10, and 12.

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Points to Remember:

  • Find the least common multiple (LCM) of 8, 10, and 12.
  • Identify the largest three-digit multiple of the LCM.

Introduction:

This question requires a mathematical approach to find the largest three-digit number divisible by 8, 10, and 12. This involves determining the least common multiple (LCM) of these three numbers and then finding the largest three-digit multiple of that LCM. The LCM represents the smallest number that is a multiple of all three given numbers.

Body:

1. Finding the Least Common Multiple (LCM):

To find the LCM of 8, 10, and 12, we first find the prime factorization of each number:

  • 8 = 2³
  • 10 = 2 × 5
  • 12 = 2² × 3

The LCM is found by taking the highest power of each prime factor present in the factorizations:

LCM(8, 10, 12) = 2³ × 3 × 5 = 8 × 3 × 5 = 120

2. Finding the Largest Three-Digit Multiple of the LCM:

Now we need to find the largest three-digit number that is divisible by 120. The largest three-digit number is 999. We divide 999 by 120 to find how many times 120 goes into 999:

999 ÷ 120 = 8 with a remainder of 39

This means that 8 times 120 (8 x 120 = 960) is the largest multiple of 120 that is less than or equal to 999.

Therefore, the largest three-digit number completely divisible by 8, 10, and 12 is 960.

Conclusion:

In summary, by finding the least common multiple of 8, 10, and 12 (which is 120), and then determining the largest three-digit multiple of this LCM, we have identified 960 as the solution. This problem highlights the importance of understanding fundamental mathematical concepts like prime factorization and least common multiples in solving seemingly complex problems. This approach can be generalized to find the largest n-digit number divisible by any set of given numbers. The solution demonstrates a clear and efficient method for solving such problems, emphasizing the power of systematic mathematical reasoning.

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