Problem 23 - 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. $$79$$

Step-by-Step Solution

Answer

79 is a prime number. It does not need to be factored because prime numbers are their own prime factors.

Step 1: Identify if the number is prime or composite

Determine if the number 79 has any divisors other than 1 and itself. This can be done by attempting to divide the number by every integer greater than 1 and less than the square root of the number. A prime number does not have any divisors other than 1 and itself. Therefore, if no other divisors are found, the number is prime.

Step 2: Calculate the prime factorization

If the number was composite, it would be broken down into a product of prime numbers. However, since 79 is a prime number, it does not need to be factored. Its only prime factor is 79 itself.

Key Concepts

Natural Numbers

The realm of numbers is quite vast, but natural numbers have a special place. These are the numbers you probably first learned to count with. Natural numbers are the set of positive integers starting from 1, going upwards: 1, 2, 3, 4, and so on. They are the numbers you can count on your fingers and toes.

Here are some cool things about natural numbers:

• Natural numbers do not include zero or any negative numbers.
• They are used for counting and ordering. For example, when lining up in a queue or ranking in a competition.
• Every natural number has an infinite number of multiples. For instance, the multiples of 2 are 2, 4, 6, 8, hinspace ext{etc}.

When it comes to prime and composite numbers, it's always these natural numbers that we're talking about. This makes understanding them fundamental as we explore deeper mathematical concepts.

Prime Factorization

Prime factorization is a method of expressing a composite number as a product of prime numbers. It’s like finding the unique building blocks of a given number.

To do prime factorization, you repeatedly divide the number by its smallest prime divisor until you're left with 1. Here's how you do it:

• Begin with the smallest prime number, which is 2.
• Divide your number by 2 if possible. If not, go to the next prime number like 3, 5, 7, etc.
• Repeat the process with the quotient until no further division is possible.

Let's say we had a composite number like 12, the prime factorization would be 2 \(\times\) 2 \(\times\) 3. That means the number 12 relies entirely on these primes. Prime factorization helps simplify expressions and find common denominators. It's a useful tool for many mathematical applications.

Composite Numbers

Composite numbers are like the opposite of prime numbers. Instead of having only two factors, composite numbers have more than two.

What does that mean? Well, composite numbers can be divided by numbers other than 1 and themselves. Take the number 4, for instance: you can divide it by 1, 2, and 4. Since there are factors other than 1 and itself, 4 is composite.

• The smallest composite number is 4 because it is the first number with factors beyond just 1 and itself.
• All even numbers greater than 2 are composite because they’re divisible by 2.
• If a number can be expressed as a product of more than one set of factors, it's composite.

Understanding composite numbers is essential, because whenever we’re trying to break down numbers into their simplest forms, we're identifying the composite numbers from which primes emerge. Without composite numbers, prime factorization wouldn't exist!