The Greatest Three-Digit Multiple of 8 and 9: A Comprehensive Guide

Understanding the greatest three-digit number that is a multiple of both 8 and 9 requires a bit of mathematical reasoning. In this article, we will explore how to find this number using the concept of the least common multiple (LCM). We will delve into the prime factorizations, the LCM calculation, and the process of determining the greatest three-digit multiple of 72.

Introduction to Finding the LCM of 8 and 9

When looking for the greatest three-digit number that is a multiple of both 8 and 9, the first task is to determine the least common multiple (LCM) of these two numbers. This LCM will help us find the multiples and ultimately the greatest one that falls within the three-digit range.

Prime Factorization of 8 and 9

Let's break down the prime factorizations of 8 and 9:

8: The prime factorization of 8 is (2^3). 9: The prime factorization of 9 is (3^2).

Calculating the LCM

The LCM is found by taking the highest power of each prime that appears in the factorizations. Therefore:

(text{LCM}(8,9) 2^3 times 3^2 8 times 9 72)

By multiplying the highest powers of the prime numbers, we get 72 as the least common multiple of 8 and 9.

Identifying the Greatest Three-Digit Multiple of 72

Now we need to find the greatest three-digit number that is a multiple of 72. To do this, we start by identifying the range of three-digit numbers and then determine the largest multiple within this range.

The greatest three-digit number is 999. We need to find the largest multiple of 72 that is less than or equal to 999.

Dividing 999 by 72

Let's start by dividing 999 by 72:

999 ÷ 72 ≈ 13.875

We take the integer part of this division, which is 13. This quotient will help us find the Greatest Common Divisor (GCD) multiple.

Now, we multiply 13 by 72:

72 × 13 936

Thus, the greatest three-digit number that is a multiple of both 8 and 9 is 936.

Generalizing for Any Base Larger Than 9

It's also worth noting that for a base larger than 9, the general formula for the greatest three-digit multiple of 72 would be:

(8 times 9 times leftlfloor frac{1000}{8 times 9} rightrfloor)

Let's break this down:

8: This is the factor from the prime factorization of 8. 9: This is the factor from the prime factorization of 9. (leftlfloor frac{1000}{72} rightrfloor): This represents the greatest integer less than or equal to (frac{1000}{72}).

Calculating the floor division:

(leftlfloor frac{1000}{72} rightrfloor leftlfloor 13.8888ldots rightrfloor 13)

Thus, the expression simplifies to:

(8 times 9 times 13 936)

So, the greatest three-digit multiple of 72 in any base larger than 9 is still 936.

Conclusion

To summarize, the greatest three-digit number that is a multiple of both 8 and 9 is 936. This was determined by calculating the least common multiple (LCM) of 8 and 9, which is 72, and then finding the largest multiple of 72 within the three-digit range. Whether in base 10 or any base larger than 9, the process remains consistent, and the answer remains 936.