Exploring the Largest 3-Digit Number Divisible by 12, 16, and 20

Understanding the Least Common Multiple (LCM) is a fundamental skill in mathematics, particularly useful in solving problems that involve divisibility. In this article, we will delve into the process of finding the largest 3-digit number divisible by 12, 16, and 20. This exploration will not only provide a clear solution but also highlight important concepts in number theory.

Introduction to Divisibility and LCM

Divisibility refers to the property of an integer being exactly divisible by another integer. The concept of the Least Common Multiple (LCM) is central to this property, as it represents the smallest positive integer that is divisible by multiple integers. In this case, the LCM of 12, 16, and 20 is the smallest number that can be evenly divided by all three numbers.

Step-by-Step Calculation of LCM

To find the LCM of the numbers 12, 16, and 20, we start by factoring each number into its prime factors:

12 22 × 3 16 24 × 1 20 22 × 5

We then take the highest power of each prime factor appearing in the factorization of any of these numbers to compute the LCM:

LCM 24 × 3 × 5 240

Understanding the Smallest and Largest Multiples

While the LCM, 240, represents the smallest positive number divisible by 12, 16, and 20, the problem asks for the largest 3-digit number with this property. To find this number, we observe the sequence of multiples of 240:

240 × 1 240 240 × 2 480 240 × 3 720 240 × 4 960

The next multiple, 240 × 5 1200, is a 4-digit number and thus outside the scope of our problem. Therefore, the largest 3-digit number divisible by 12, 16, and 20 is:

960

Confusion and Common Mistakes

Many might believe that the answer is simply 240, since it is the LCM. However, as we calculated, 240 is not a 3-digit number. The correct approach is to find the largest 3-digit multiple of 240, which is 960. This highlights the importance of understanding the question fully and not being misled by simple misconceptions.

Conclusion

In summary, finding the largest 3-digit number divisible by 12, 16, and 20 involves identifying the LCM of these numbers, which is 240, and then finding the largest 3-digit multiple of this LCM. The answer is 960. This problem emphasizes the significance of prime factorization, LCM, and careful interpretation of the question.

Related Keywords

Largest 3-digit number Divisibility LCM

Further Reading

For a deeper understanding of LCM, divisibility, and number theory, you can explore these resources:

Khan Academy: Least Common Multiple Example Math Is Fun: Factors and LCM Math Is Fun: Divisibility Rules