Problem 19 - 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. $$36$$

Step-by-Step Solution

Answer

The number 36 is composite and its prime factorization is \(2^2*3^2\).

Step 1: Determine if the number is prime

First, you need to check if the number \(36\) has exactly two distinct divisors. This can be easily done since \(36\) can be obtained by multiplying two numbers other than 1 and itself (for example \(6*6\) or \(4*9\)), so it is not a prime number.

Step 2: Prime Factorization

Since the number \(36\) is not a prime number, it’s necessary to find its prime factorization. You can do this by dividing \(36\) by the smallest prime number (which is 2) and continue this process until the result is a prime number. Doing so, you will find that \(36 = 2*2*3*3\).

Step 3: Write the result

The prime factorization of a number is usually written by using exponents, so we write \(36 = 2^2*3^2\).

Key Concepts

Natural Numbers

Natural numbers are one of the simplest concepts in mathematics and are usually the numbers you learn to count with, starting from 1. They are defined as the set of positive integers starting from 1, extending infinitely upward (i.e., 1, 2, 3, 4,...).

• Natural numbers include numbers like 1, 2, 3, and so on.
• They do not include zero in their basic definition (although some mathematicians include zero, this is more common in the definition of whole numbers).
• They are always positive and do not include fractions or decimals.

These numbers are straightforward because they only involve basic properties such as addition and multiplication. It's important to know them well as they are used to build up more complex mathematical concepts.

Composite Numbers

A composite number is a natural number greater than 1 that is not prime. This means it has more than two distinct positive divisors. In simpler terms, a composite number can be divided by numbers other than 1 and itself without leaving a remainder.

• The number 4 is an example of a composite number because it can be divided by 1, 2, and 4.
• 6, 8, 9, 10, and so on, are also composite numbers.
• Every composite number can be expressed as a product of prime numbers, known as its prime factorization.

For example, in the original exercise, the number 36 was identified as composite. By factorizing it into prime numbers, we find that it equals \(2^2 \times 3^2\). Having a clear understanding of composite numbers will help you determine the factors of a number effortlessly.

Prime Numbers

Prime numbers are like the building blocks of natural numbers. These are natural numbers greater than 1 that have no divisors other than 1 and themselves. Put simply, they can only be divided evenly (without a remainder) by 1 and the number itself.

• Examples of prime numbers include 2, 3, 5, 7, and 11. Notice that 2 is the only even prime number. Every other even number can be divided by 2 and hence is composite.
• Prime numbers are crucial in mathematics, especially in number theory, because every natural number greater than 1 is either a prime or can be factored into primes.
• The process of finding which prime numbers multiply together to make a given composite number is called prime factorization.

Recognizing prime numbers and understanding their properties is essential for many areas of mathematics, and they form the backbone for various algorithms in computer science, especially in cryptography.