The Least Common Multiple: Finding the Smallest Number Divisible by 2 to 11

Introduction

The least common multiple (LCM) is a concept in number theory that helps us find the smallest number that is divisible by a set of given numbers. In this article, we will delve into how to calculate the LCM for the numbers 2 through 11, illustrating the process using the example of finding the smallest number divisible by all these integers.

Understanding the Concept of LCM

The least common multiple of a set of integers is the smallest positive integer that is divisible by each of them without leaving a remainder. It is a fundamental concept in arithmetic and is widely used in various mathematical applications such as fractions, algebra, and number theory.

Prime Factorization: The Key to Finding LCM

To find the LCM of the numbers 2 through 11, we start by determining their prime factorizations:

2: 21 3: 31 4: 22 5: 51 6: 21 times; 31 7: 71 8: 23 9: 32 10: 21 times; 51 11: 111

Next, we identify the highest powers of each prime factor that appear in these factorizations. This involves selecting the maximum exponent for each prime number derived from the factorizations.

Calculating the LCM of 2 to 11

The highest powers of the prime factors are as follows:

2: 23 (from 8) 3: 32 (from 9) 5: 51 (from 5 or 10) 7: 71 (from 7) 11: 111 (from 11)

Now, we can calculate the LCM by multiplying these highest powers together:

[ text{LCM} 2^3 times 3^2 times 5 times 7 times 11 ]

Let's break this down step-by-step:

23 8 32 9 5 5 7 7 11 11

Now, we multiply these together:

[ 8 times 9 72 ]

[ 72 times 5 360 ]

[ 360 times 7 2520 ]

[ 2520 times 11 27720 ]

Thus, the smallest number that can be divided by all the numbers from 2 to 11 is 27,720.

Considering Negative Numbers

If negative numbers are allowed, then there is no such least number. However, the least non-negative number that can be divided by these numbers is zero. The least positive number that can be divided by all the numbers from 2 to 11 is 27,720.

Some might ask, What if we consider the smallest positive integer that can be divided evenly by all these numbers? The answer remains 27,720, as it is the smallest number that meets the criteria for divisibility.

Summary and Conclusion

In this article, we have explored the process of finding the least common multiple of the numbers 2 through 11. By using prime factorization and identifying the highest powers of each prime factor, we determined that the smallest number divisible by all these numbers is 27,720. Understanding the LCM is crucial for solving various mathematical problems and applications, making it a valuable concept in the field of number theory.