key term - Least Common Multiple

Definition

The least common multiple (LCM) is the smallest positive integer that is divisible by two or more given integers without a remainder. It is a fundamental concept in mathematics that is crucial for operations involving fractions and rational expressions.

5 Must Know Facts For Your Next Test

1. The LCM is used to find a common denominator when adding or subtracting fractions with different denominators.
2. To find the LCM of two or more numbers, you can use the prime factorization method or the listing method.
3. The LCM is always greater than or equal to the product of the numbers, and it is the smallest positive integer with this property.
4. The LCM and the GCD of two or more numbers are related by the formula: $\text{LCM}(a, b) \times \text{GCD}(a, b) = a \times b$.
5. Knowing the LCM is essential for adding and subtracting rational expressions with a common denominator in Algebra.

Review Questions

• Explain how the least common multiple (LCM) is used in the context of adding and subtracting fractions with different denominators.
  • When adding or subtracting fractions with different denominators, it is necessary to find a common denominator to perform the operation. The least common multiple (LCM) of the denominators is the smallest positive integer that is divisible by all the denominators without a remainder. By finding the LCM, you can convert the fractions to equivalent fractions with the same denominator, allowing you to add or subtract the numerators directly.
• Describe the relationship between the least common multiple (LCM) and the greatest common divisor (GCD) of two or more numbers.
  • The least common multiple (LCM) and the greatest common divisor (GCD) of two or more numbers are related by the formula: $\text{LCM}(a, b) \times \text{GCD}(a, b) = a \times b$. This means that the product of the LCM and the GCD of two numbers is equal to the product of the numbers themselves. Understanding this relationship is crucial in applications involving fractions and rational expressions, as the LCM and GCD are often used together to simplify and manipulate these mathematical objects.
• Explain how the concept of the least common multiple (LCM) is applied in the context of adding and subtracting rational expressions with a common denominator.
  • In the context of adding and subtracting rational expressions with a common denominator, the least common multiple (LCM) plays a crucial role. To add or subtract rational expressions, the denominators must be the same. The LCM of the denominators is the smallest positive integer that is divisible by all the denominators without a remainder. By finding the LCM, you can convert the rational expressions to equivalent expressions with the same denominator, allowing you to perform the addition or subtraction operation on the numerators directly. This application of the LCM is essential in simplifying and manipulating rational expressions in Algebra.

Related terms

Common Multiple: A common multiple is a number that is divisible by two or more given integers.

Prime Factorization:

The process of expressing a number as a product of its prime factors.

Greatest Common Divisor (GCD): The largest positive integer that divides two or more integers without a remainder.