Multiples are the products of a given whole number and any other whole number. They represent the numbers that can be divided evenly by the original whole number without a remainder.
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The first multiple of a number is the number itself, and subsequent multiples are found by repeatedly adding the original number.
Multiples of a number are always divisible by that number without a remainder.
The set of multiples of a number is infinite, as you can continue to add the original number to find more multiples.
Multiples can be used to identify common factors between two or more numbers.
Identifying the multiples of a number is an important skill in understanding number relationships and properties.
Review Questions
Explain how multiples are related to factors and provide an example.
Multiples and factors are closely related concepts. Factors are the numbers that can be multiplied together to produce a given whole number, while multiples are the products of a given whole number and any other whole number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, and the multiples of 12 include 12, 24, 36, 48, and so on. The multiples of a number are always divisible by the factors of that number.
Describe the relationship between prime numbers, composite numbers, and multiples.
Prime numbers are whole numbers greater than 1 that have no positive divisors other than 1 and themselves. Composite numbers are whole numbers greater than 1 that have at least one positive divisor other than 1 or the number itself. The multiples of a prime number are only the prime number itself and the products of the prime number and other whole numbers. In contrast, the multiples of a composite number include the multiples of its prime factors. Understanding the relationships between these concepts is crucial for identifying patterns and properties of numbers.
Analyze how the concept of multiples can be used to solve problems related to divisibility and common factors.
The concept of multiples can be applied to solve problems related to divisibility and finding common factors between numbers. For example, if you need to determine if a number is divisible by another number, you can check if the first number is a multiple of the second number. Additionally, the common multiples of two or more numbers can be used to identify their greatest common factor. By finding the smallest positive number that is a multiple of all the given numbers, you can determine their greatest common factor. This understanding of multiples is essential for solving a variety of problems in elementary algebra and number theory.