Multiples are the products of a given number and any whole number. They represent the set of numbers that can be divided evenly by the original number without a remainder.
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Multiples of a number can be found by repeatedly multiplying the number by successive whole numbers.
The set of multiples of a number is infinite, as the number can be multiplied by any positive integer.
Identifying multiples is important in understanding divisibility, factors, and common multiples.
Multiples can be used to solve problems involving fractions, ratios, and proportions.
Finding the least common multiple (LCM) of two or more numbers is a useful skill in algebra and number theory.
Review Questions
Explain how multiples are related to the concept of divisibility.
Multiples are closely tied to the concept of divisibility because a number is a multiple of another number if and only if it is divisible by that number without a remainder. For example, 12 is a multiple of 3 because 12 is divisible by 3 with no remainder. Identifying multiples can help determine if one number is divisible by another, which is an important skill in algebra and number theory.
Describe the relationship between multiples and factors.
Multiples and factors are inverse operations. While factors are the numbers that divide evenly into another number, multiples are the products of a number and any whole number. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12, and the multiples of 3 are 3, 6, 9, 12, 15, and so on. Understanding the relationship between multiples and factors is crucial for solving problems involving divisibility, fractions, and other algebraic concepts.
Analyze the importance of finding the least common multiple (LCM) of two or more numbers.
The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more given numbers. Finding the LCM is essential in many algebraic applications, such as simplifying fractions, solving equations with fractions, and working with ratios and proportions. The LCM allows you to find a common denominator for fractions, which is necessary for performing operations like addition and subtraction. Additionally, the LCM is used in number theory to understand the relationships between different numbers and their divisibility properties.