A multiple is a number that can be expressed as a product of a given number and a whole number. In other words, a multiple is the result of multiplying a number by a positive integer.
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Multiples are important in the context of finding the Greatest Common Factor (GCF) and factoring by grouping.
The GCF of two or more numbers is the largest positive integer that divides each of the numbers without a remainder.
To find the GCF, you need to identify the common multiples of the given numbers.
Factoring by grouping involves identifying the common factors among groups of terms and then factoring out the common factor.
Multiples are essential in this process because they help identify the common factors that can be factored out.
Review Questions
Explain how the concept of multiples is used in the process of finding the Greatest Common Factor (GCF) of two or more numbers.
The concept of multiples is crucial in finding the GCF of two or more numbers. To find the GCF, you need to identify the common multiples of the given numbers. This involves listing the multiples of each number and then finding the largest positive integer that is a common multiple of all the given numbers. The GCF is the largest positive integer that divides each of the numbers without a remainder, and it can be found by identifying the common multiples of the given numbers.
Describe the role of multiples in the process of factoring by grouping.
Multiples play a significant role in the process of factoring by grouping. Factoring by grouping involves identifying the common factors among groups of terms and then factoring out the common factor. To identify the common factors, you need to look for the common multiples of the coefficients or variables in the given expression. By identifying the common multiples, you can then factor out the common factor, which is the largest positive integer that divides each of the terms without a remainder. The concept of multiples is essential in this process, as it helps you recognize the common factors that can be factored out.
Analyze how the understanding of multiples can help you solve problems related to the Greatest Common Factor (GCF) and factoring by grouping.
A deep understanding of multiples is crucial for solving problems related to the GCF and factoring by grouping. By recognizing multiples, you can identify the common factors among the given numbers or terms, which is the key step in finding the GCF and factoring by grouping. Specifically, listing the multiples of each number or term and finding the common multiples allows you to determine the GCF. Similarly, identifying the common multiples of the coefficients or variables in an expression enables you to factor out the common factor through the process of factoring by grouping. This understanding of multiples and their relationship to common factors is essential for successfully solving problems involving the GCF and factoring by grouping.