Least Common Multiple (LCM)

Definition

The least common multiple (LCM) is the smallest positive integer that is divisible by two or more given integers. It is a fundamental concept in algebra that is used to solve various types of problems, including linear equations and the division of radical expressions.

5 Must Know Facts For Your Next Test

1. The LCM of two or more integers is the smallest positive integer that is divisible by all the given integers.
2. The LCM can be found by first finding the prime factorization of each integer, and then taking the product of the highest power of each prime factor.
3. The LCM is often used to find a common denominator when adding or subtracting fractions with different denominators.
4. In the context of solving linear equations, the LCM is used to eliminate fractions and simplify the equation.
5. When dividing radical expressions, the LCM is used to find a common denominator, which is necessary for the division process.

Review Questions

• Explain how the concept of the least common multiple (LCM) is used in the context of solving linear equations.
  • In the context of solving linear equations, the LCM is used to eliminate fractions and simplify the equation. By finding the LCM of the denominators of the fractions in the equation, you can then multiply both sides of the equation by the LCM to get rid of the fractions. This allows you to solve the equation using standard algebraic methods, as the LCM ensures that all the terms have a common denominator.
• Describe how the LCM is used when dividing radical expressions.
  • When dividing radical expressions, the LCM is used to find a common denominator, which is necessary for the division process. By finding the LCM of the radicands (the numbers under the radical signs), you can then rewrite the expressions with a common denominator. This allows you to perform the division operation by dividing the numerators and simplifying the resulting expression.
• Analyze the relationship between the LCM and the GCF, and explain how understanding this relationship can be useful when working with algebraic expressions.
  • The LCM and the GCF are closely related concepts, as they both involve the divisibility of integers. The relationship between the LCM and the GCF is that their product is equal to the product of the given integers. This means that if you know the GCF of two or more integers, you can use it to find the LCM, and vice versa. Understanding this relationship can be useful when working with algebraic expressions, as it allows you to simplify and manipulate expressions more efficiently by identifying common factors or multiples.

Related terms

Greatest Common Factor (GCF): The greatest common factor (GCF) is the largest positive integer that divides each of the given integers without a remainder. The GCF and LCM are closely related concepts.

Prime Factorization: Prime factorization is the process of expressing a positive integer as a product of prime numbers. It is a useful tool for finding the LCM of two or more integers.

Divisibility: Divisibility is the property of one number being evenly divisible by another number, without leaving a remainder. Understanding divisibility is crucial for finding the LCM.