What is the Largest 5-Digit Number Divisible by 4, 5, 6, 7, 8, 9, 10, 11?

In this article, we will explore the mathematical process involved in determining the largest 5-digit number that is divisible by all the given numbers: 4, 5, 6, 7, 8, 9, 10, and 11. This problem involves understanding divisibility rules and the concept of Least Common Multiple (LCM). We will break down the steps involved, explain the calculations, and provide a detailed walkthrough to help our users understand the solution better.

Understanding Divisibility

First, let's briefly understand what it means for a number to be divisible by another. For example:

Divisible by 4: The number must be even and the last two digits form a number that is divisible by 4. Divisible by 5: The last digit must be either 0 or 5. Divisible by 6: The number must be divisible by both 2 and 3. Divisible by 7: There is a specific rule for 7 that is simpler to work through directly. Divisible by 8: The last three digits form a number that is divisible by 8. Divisible by 9: The sum of the digits must be divisible by 9. Divisible by 10: The last digit must be 0. Divisible by 11: The difference between the sum of the digits in the odd positions and the sum of the digits in the even positions must be divisible by 11.

Calculating the Least Common Multiple (LCM)

To find the LCM of the numbers 4, 5, 6, 7, 8, 9, 10, and 11, we need to find the product of the highest powers of all prime factors present. Here are the prime factorizations of each number:

4 22 5 prime 6 2 × 3 7 prime 8 23 9 32 10 2 × 5 11 prime

The LCM is the product of the highest powers of all these primes:

LCM 23 × 32 × 5 × 7 × 11 27,720

Applying the Result

Since we are looking for the largest 5-digit number divisible by all the given numbers, we know that 27,720 itself is already a 5-digit number. This means the largest such number is simply 3 times 27,720, because after 27,720, the next multiple would be 27,720 × 4, which is a 6-digit number.

Thus, the largest 5-digit number divisible by 4, 5, 6, 7, 8, 9, 10, and 11 is:

27,720 × 3 83,160

Conclusion

With this solution, we have demonstrated a method to find the largest 5-digit number divisible by several given numbers. The process involves understanding basic divisibility rules and calculating the LCM of the numbers. This technique is useful not only in mathematical problems but also in various practical applications, such as planning and scheduling.

Understanding these concepts can be valuable for SEO optimization as well, especially in content related to mathematics, rules, and problem-solving techniques.