Question

Explain how to check a prime factorization.

   Explain how to check a prime factorization.

Prealgebra

Alan S. Tussy, R.… 4th Edition

Chapter 1, Problem 105

Instant Answer

Solved by Expert Amy Jiang

07/18/2020

Step 1

Prime factors are the prime numbers that divide the number exactly. For example, let's say we have the number 18. The prime factors of 18 are 2 and 3. So, the prime factorization of 18 is \(2 \times 3 \times 3\). Show more…

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Explain how to check a prime factorization.

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Key Concepts

Prime Factorization

Prime factorization is the process of expressing a composite number as a product of its prime numbers. This breakdown illustrates exactly which prime numbers multiply together to reconstruct the original number. It serves as a fundamental tool in number theory and various applications in mathematics, such as simplifying fractions or solving problems involving divisibility.

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely written as a product of prime numbers, apart from the order of the factors. This concept underpins the validity of prime factorization and ensures that once a composite number is decomposed into primes, there's no alternative set of prime factors that can produce the same number.

Prime Numbers

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. These numbers serve as the basic building blocks for all natural numbers when represented through multiplication, making them central to the study of integer properties and prime factorization.

Verification Process

To check a prime factorization, one multiplies all the prime factors together to see if the product equals the original composite number. Additionally, it’s important to verify that each factor in the decomposition is indeed a prime number. This two-step process – confirming both the multiplication result and the primality of factors – ensures the correctness of the reported factorization.

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