Introduction

Finding the least common multiple (LCM) is a fundamental concept in mathematics that has numerous applications, from simplifying fractions to solving problems in number theory. This article examines the smallest number that has the factors 1, 2, 3, 4, 5, 6, 7, 8, and 9. We will use prime factorization and the least common multiple to solve this problem and explain the mathematical principles behind it.

The Least Common Multiple (LCM)

The least common multiple of a set of integers is the smallest positive integer that is divisible by each of the integers in the set. For the numbers 1 through 9, the LCM can be determined through prime factorization and multiplication of the highest powers of each prime.

Prime Factorization

First, let's break down each number into its prime factors: t1: No prime factors t2: 21 t3: 31 t4: 22 t5: 51 t6: 21 × 31 t7: 71 t8: 23 t9: 32 Next, we determine the highest power of each prime that appears in these factorizations:
tThe highest power of 2 is 23 from 8. tThe highest power of 3 is 32 from 9. tThe highest power of 5 is 51 from 5. tThe highest power of 7 is 71 from 7.

Calculating the LCM

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

LCM 23 × 32 × 51 × 71

Let's break this down step by step: t23 8 t32 9 t51 5 t71 7 t8 × 9 72 t72 × 5 360 t360 × 7 2520 Thus, the smallest number that has the factors 1, 2, 3, 4, 5, 6, 7, 8, and 9 is 2520.

Verification

To verify, we can use the J programming language. Using the LCM verb(/): ./1 to 10

The result is 2520.

We can also check by dividing 2520 by the factors: t2520 / 9 280, which has factors 9, 3, and 1. t280 / 7 40, which has factor 7. t40 / 8 5, which has factors 8, 5, 4, and 2. t

Since 2520 has both 2 and 3 as factors, it also has 6 as a factor.

Conclusion

The least common multiple (LCM) is a powerful tool in mathematics, and its understanding and application can help solve complex number theory problems. The process of finding the LCM through prime factorization is both logical and straightforward, making it an essential concept in any mathematical toolkit.

Keywords

tLeast Common Multiple (LCM) tPrime Factorization tFactors