A multiple is a number that can be divided by another number without a remainder. Multiples are an important concept in mathematics, particularly in the context of dividing whole numbers and understanding prime factorization and the least common multiple.
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Multiples of a number are all the numbers that can be obtained by multiplying that number by an integer.
When dividing whole numbers, the quotient is a multiple of the divisor, and the remainder is a factor of the divisor.
In prime factorization, the multiples of a number are found by multiplying its prime factors together in different combinations.
The least common multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the given numbers.
Knowing the multiples of a number is essential for understanding concepts like divisibility, prime factorization, and the least common multiple.
Review Questions
Explain how the concept of multiples is used in the division of whole numbers.
When dividing whole numbers, the quotient is a multiple of the divisor. For example, if we divide 24 by 6, the quotient is 4, which is a multiple of 6 (4 × 6 = 24). The remainder, if any, is a factor of the divisor. Understanding multiples in the context of division helps students recognize the relationship between the divisor, quotient, and remainder, and develop a deeper understanding of the division process.
Describe how the concept of multiples is used in the process of prime factorization.
In prime factorization, the multiples of a number are found by multiplying its prime factors together in different combinations. For instance, to find the multiples of 12, we first identify its prime factors as 2 and 3. The multiples of 12 are then 12, 6, 4, 3, 2, and 1, which are obtained by multiplying these prime factors in various ways. Recognizing the role of multiples in prime factorization helps students understand the underlying structure of numbers and how to efficiently break them down into their prime components.
Explain the relationship between multiples and the least common multiple (LCM) of two or more numbers.
The least common multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the given numbers. To find the LCM, students need to identify the multiples of each number and then determine the smallest number that is a multiple of all the given numbers. Understanding the concept of multiples is crucial for finding the LCM, as it allows students to recognize the relationships between the numbers and identify the common multiples that will lead to the smallest positive integer that is a multiple of all the numbers.