Problem 134 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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 \(2^2 \times 3^2\).