Composite numbers are positive integers that have at least one positive divisor other than 1 or the number itself. They are the opposite of prime numbers, which have no positive divisors other than 1 and the number itself.
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Composite numbers can be written as a product of two or more prime numbers.
The prime factorization of a composite number reveals its prime factors and their multiplicities.
The least common multiple of two or more composite numbers can be found by multiplying their prime factors and their highest multiplicities.
Identifying composite numbers is important for understanding prime factorization and calculating the least common multiple.
Composite numbers are the building blocks for many mathematical concepts, such as the Fundamental Theorem of Arithmetic.
Review Questions
Explain how the concept of composite numbers is related to prime factorization.
Composite numbers are defined as positive integers that have at least one positive divisor other than 1 or the number itself. This means that a composite number can be expressed as a product of two or more prime numbers, which is the process of prime factorization. By finding the prime factors of a composite number, you can determine its divisors and use this information to perform operations like finding the least common multiple.
Describe the relationship between composite numbers and the least common multiple (LCM).
The least common multiple of two or more integers is the smallest positive integer that is divisible by all of the given integers. Since composite numbers can be expressed as a product of prime factors, the LCM of composite numbers can be found by multiplying the prime factors and their highest multiplicities. This is because the prime factors of the composite numbers, and their respective powers, determine the divisibility relationships between the numbers, which is the key to finding the LCM.
Analyze how the concept of composite numbers is fundamental to the Fundamental Theorem of Arithmetic.
The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be written as a unique product of prime numbers. This theorem is built upon the understanding of composite numbers, which are positive integers that have at least one positive divisor other than 1 or the number itself. Composite numbers can be expressed as a product of their prime factors, and this prime factorization is unique, which is the core of the Fundamental Theorem of Arithmetic. The properties of composite numbers, including their divisibility relationships and prime factorizations, are essential to the proof and application of this fundamental theorem in number theory.