Problem 18 - Introductory and Intermediate Algebra for College Students 4th

Question

Identify each natural number as prime or composite. If the number is composite, find its prime factorization. $$23$$

Step-by-Step Solution

Answer

The number 23 is a prime number and prime numbers do not have a prime factorization as they can only be divided evenly by 1 and themselves.

Step 1: Determine if the number is prime or composite

Check if the number can be divided evenly by any numbers other than 1 and itself. In this case, 23 cannot be divided evenly by any other number, so 23 is a prime number.

Step 2: Find prime factorization

This step should be performed only if the number is found to be composite in the first step. As in this case, the number is prime, there is no need to carry out this step.

Key Concepts

Understanding Natural Numbers

When we talk about natural numbers, we're referring to the set of numbers that we use for counting and ordering in everyday life. Starting from the number 1 and moving upwards, every whole number that follows falls under this category, including 2, 3, 4, and so on. They are the simplest type of numbers and do not include fractions, decimals, or negative numbers.

Whenever you're working with mathematical problems like identifying whether a natural number is prime or composite, you begin by considering these counting numbers. For instance, take the number 23 from the exercise. It's a part of the natural numbers set, and a fundamental property we investigate is whether it is prime or composite.

Deciphering Prime Numbers

Prime numbers are the 'building blocks' of natural numbers. Think of them as the original, indivisible numbers that you cannot break down into simpler natural number factors. Specifically, a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

In other words, if you cannot divide a number evenly (without leaving a remainder) by any other number besides 1 and the number itself, it is prime. This is exactly what we see in the exercise with the number 23. It does not have any divisors other than 1 and 23, so we categorize it as a prime number. Remember that prime numbers can only be found among the natural numbers, and their key characteristic is indivisibility by any other natural number except for 1.

Identifying Composite Numbers

Composite numbers contrast prime numbers. These numbers are also part of the natural numbers set but what makes them different is that they can be divided by at least one other natural number besides 1 and themselves. They are essentially the 'team players' of natural numbers, allowing for multiplication combinations that result in themselves.

For example, let's take the number 8. It can be divided by 1, 2, 4, and 8, making it composite. When dealing with a composite number, one of the tasks could be to perform prime factorization, which is the process of expressing the number as a product of prime numbers. However, since the number 23 from the exercise is prime and not composite, it does not have a prime factorization.