Finding the Least Common Multiple (LCM) of Consecutive Integers x, x-1, and x 1

When dealing with the mathematical concept of finding the least common multiple (LCM) of algebraic expressions, one common scenario involves consecutive integers. In this article, we will explore how to find the LCM of the expressions x, x-1, and x 1. We will break down the process step-by-step to ensure a thorough understanding of the concept.

Understanding the Expressions

The expressions x, x-1, and x 1 represent three consecutive integers. Consecutive integers are a sequence of integers where each number is one more than the previous one. For example, if x 2, then the consecutive integers are 1, 2, and 3.

Properties of Consecutive Integers

One important property of consecutive integers is that their LCM is equal to their product. This is due to the fact that each integer in the set is relatively prime to the others, meaning they share no common factors other than 1.

LCM of Three Consecutive Integers

The LCM of any three consecutive integers can be expressed using the formula:

LCM(x, x-1, x 1) x(x-1(x 1) / gcd(x, x-1, x 1)

The greatest common divisor (GCD) of three consecutive integers is always 1. This is because consecutive integers are coprime, meaning they share no common factors other than 1.

Conclusion of the LCM Calculation

Given the above properties, the LCM of the expressions x, x-1, and x 1 can be simplified to:

LCM(x, x-1, x 1) x(x-1(x 1) x(x2 - 1)

Therefore, the LCM of x, x-1, and x 1 is:

LCM(x, x-1, x 1) xx2 - 1

Conclusion

The least common multiple of the expressions x, x-1, and x 1 is x(x2 - 1). This can be verified by recognizing that it includes all unique prime factors and that the GCD of the three expressions is 1.

Example:

If x 2, then the consecutive integers are 1, 2, and 3. The LCM of 1, 2, and 3 is 6. Using our formula:

LCM(2, 1, 3) 2(22 - 1) 2(4 -1) 2(3) 6

This verifies our formula as correct.