Product Of Digits Largest 4-Digit Number Divisible By 12 18 And 27

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Finding the product of digits of a number that meets specific divisibility criteria is a fascinating mathematical problem. In this article, we will explore how to determine the largest 4-digit number that is divisible by 12, 18, and 27, and then calculate the product of its digits. This problem combines concepts from number theory, including divisibility rules and finding the least common multiple (LCM). By working through this problem, we can enhance our understanding of these fundamental mathematical principles.

Understanding Divisibility and the Least Common Multiple (LCM)

To solve this problem, it's crucial to grasp the concepts of divisibility and the least common multiple (LCM). A number is divisible by another if the remainder is zero after division. For example, 24 is divisible by 12 because 24 ÷ 12 = 2 with no remainder. Understanding the rules of divisibility for numbers like 2, 3, 4, 9, etc., can greatly simplify the process of identifying multiples.

The Least Common Multiple (LCM) of a set of numbers is the smallest positive integer that is divisible by all the numbers in the set. The LCM is essential when we need to find a number that satisfies multiple divisibility conditions. To find the LCM, we can use prime factorization. Prime factorization involves breaking down each number into its prime factors. For instance, the prime factorization of 12 is 2 × 2 × 3 (or 2^2 × 3), of 18 is 2 × 3 × 3 (or 2 × 3^2), and of 27 is 3 × 3 × 3 (or 3^3). The LCM is then found by taking the highest power of each prime factor that appears in any of the factorizations and multiplying them together.

In our case, to find a number divisible by 12, 18, and 27, we need to determine the LCM of these three numbers. This will give us the smallest number divisible by all three, and from there, we can find larger multiples, specifically the largest 4-digit multiple.

Calculating the LCM of 12, 18, and 27

To calculate the LCM of 12, 18, and 27, we first find the prime factorization of each number:

  • 12 = 2^2 × 3
  • 18 = 2 × 3^2
  • 27 = 3^3

Next, we identify the highest power of each prime factor present in the factorizations:

  • The highest power of 2 is 2^2.
  • The highest power of 3 is 3^3.

Now, we multiply these highest powers together to get the LCM:

LCM (12, 18, 27) = 2^2 × 3^3 = 4 × 27 = 108

Therefore, the least common multiple of 12, 18, and 27 is 108. This means that any number divisible by 12, 18, and 27 must also be divisible by 108. This understanding is crucial for finding the largest 4-digit number that meets our criteria. We now need to find the largest 4-digit multiple of 108.

Finding the Largest 4-Digit Number Divisible by 108

Now that we know the LCM of 12, 18, and 27 is 108, our next step is to find the largest 4-digit number that is divisible by 108. The largest 4-digit number is 9999. To find the largest 4-digit multiple of 108, we will divide 9999 by 108 and consider the quotient.

Dividing 9999 by 108 gives us:

9999 ÷ 108 ≈ 92.583

Since we need a whole number multiple of 108, we take the integer part of the quotient, which is 92. This tells us that 108 multiplied by 92 will be the largest multiple of 108 that is less than or equal to 9999.

Now, we multiply 108 by 92:

108 × 92 = 9936

So, 9936 is the largest 4-digit number that is divisible by 108, and consequently, it is also divisible by 12, 18, and 27. This is because 108 is the LCM of these three numbers. We have successfully identified the number we need for the next part of our problem: calculating the product of its digits.

Verifying Divisibility

To ensure that 9936 is indeed divisible by 12, 18, and 27, we can perform the divisions:

  • 9936 ÷ 12 = 828
  • 9936 ÷ 18 = 552
  • 9936 ÷ 27 = 368

Since all divisions result in whole numbers, we can confirm that 9936 is divisible by 12, 18, and 27. This verification step is important to ensure accuracy before moving on to the final calculation.

Calculating the Product of the Digits

Now that we have found the largest 4-digit number divisible by 12, 18, and 27, which is 9936, we can proceed to calculate the product of its digits. The digits of 9936 are 9, 9, 3, and 6. To find the product, we simply multiply these digits together.

The product of the digits is:

9 × 9 × 3 × 6

Let's break down the multiplication:

  • 9 × 9 = 81
  • 3 × 6 = 18

Now, multiply the results:

81 × 18 = 1458

Therefore, the product of the digits of the largest 4-digit number divisible by 12, 18, and 27 is 1458. This completes the solution to our problem.

Alternative Calculation Methods

While we have shown one method to calculate the product, it’s worth noting that there are alternative ways to perform the multiplication. For example, one could multiply the numbers in a different order:

  • 9 × 3 = 27
  • 9 × 6 = 54
  • 27 × 54 = 1458

Regardless of the method used, the result remains the same. The key is to ensure accuracy in each step of the multiplication process.

Conclusion

In this article, we solved a problem that required us to find the largest 4-digit number divisible by 12, 18, and 27, and then calculate the product of its digits. We began by understanding the concepts of divisibility and the least common multiple (LCM). We calculated the LCM of 12, 18, and 27 to be 108. Then, we found the largest 4-digit multiple of 108, which is 9936. Finally, we multiplied the digits of 9936 together to get the product 1458.

This problem illustrates how different mathematical concepts can be combined to solve a single question. Understanding divisibility rules, prime factorization, and LCM is essential for tackling such problems. Moreover, this exercise enhances our problem-solving skills and reinforces our understanding of fundamental mathematical principles.

By working through this example, we hope to have provided a clear and comprehensive explanation of the steps involved in solving this type of problem. Whether you are a student learning about number theory or simply someone who enjoys mathematical challenges, this problem offers a valuable exercise in logical thinking and calculation.

Summary of Key Steps

  1. Understand Divisibility and LCM: Grasp the concepts of divisibility and the least common multiple (LCM).
  2. Calculate the LCM: Find the LCM of 12, 18, and 27 using prime factorization.
  3. Find the Largest 4-Digit Multiple: Determine the largest 4-digit number divisible by the LCM.
  4. Calculate the Product of Digits: Multiply the digits of the number obtained in the previous step.
  5. Verify the Solution: Ensure all calculations are accurate and the solution meets the initial criteria.

Practice Problems

To further reinforce your understanding, try solving similar problems:

  1. Find the product of the digits of the smallest 5-digit number divisible by 15, 20, and 25.
  2. What is the product of the digits of the largest 3-digit number divisible by 8, 12, and 16?
  3. Calculate the product of the digits of the smallest 4-digit number divisible by 9, 15, and 21.

By practicing these problems, you can solidify your understanding of the concepts discussed and improve your problem-solving skills in number theory.