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Chapter 1: Problem 134

What is meant by the prime factorization of a composite number?

Short Answer

Expert verified
The prime factorization of a composite number means expressing the composite number as a product of prime numbers. For example, the prime factorization of 36 is \(2^2 \times 3^2\).

Step by step solution

01

Understanding Composite Numbers

A composite number is a positive integer that has more than two factors. In other words, a composite number can be divided evenly at least by 1, itself, and embedded numbers. An example of a composite number is 4, since it can be divided evenly by 1, 2 and 4.
02

Understanding Primes

A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. The first six prime numbers are 2, 3, 5, 7, 11, and 13.
03

Prime Factorization

The prime factorization of a composite number is the expression of that number as a product of prime numbers. This entails breaking down a composite number into 'factors' until all the factors are prime numbers.
04

Example of Prime Factorization

For example, let's take the composite number 36. This number can be divided by 2, which yields 18. The number 18 can further be divided by 2 to give 9. Lastly, the number 9 can be divided by 3 to give 3. So, the prime factorization of 36 becomes \(2 \times 2 \times 3 \times 3\) which is commonly written in exponential form as \(2^2 \times 3^2\).

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