Problem 14 - 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. $$15$$

Step-by-Step Solution

Answer

The number 15 is a composite number. Its prime factorization is \(3 \times 5\).

Step 1: Identify whether the number is prime or composite

Starting with 2, check to see if 15 can be divided evenly by 2. If it can, then it's composite. If none of the numbers 2 through 15 (excluding 15) can divide 15 evenly, then it's prime. In this case, 15 is divisible by 3 and 5, so it's composite.

Step 2: Find prime factorization

To find the prime factorization of a composite number, divide the number by the smallest prime number (i.e., 2). If the number can be evenly divided, write down the prime number and the result of the division. If it cannot be divided evenly by 2, try the next prime number (i.e., 3), and so on. Continue this process until the result of the division is a prime number.\n\nFor 15, the smallest prime number it can be evenly divided by is 3. Dividing 15 by 3 gives 5, which is a prime number. Therefore, 15 equals 3 times 5, or \(3 \times 5\). So, the prime factorization of 15 is \(3 \times 5\)

Key Concepts

Composite Numbers

Understanding composite numbers is essential for mastering prime factorization. A composite number is a positive integer greater than one that is not prime; in other words, it has more than two distinct positive divisors.

For instance, let's look at the number 15 from our exercise. When determining whether 15 is a prime or composite number, we find that it has divisors other than 1 and itself—it can be divided evenly by 3 and 5. This makes it a composite number. Most natural numbers we encounter are, in fact, composite.

Identifying Composite Numbers

You can identify a composite number by checking if it can be divided evenly by any prime number smaller than itself. This process is sometimes referred to as prime factorization. Through this method, we discover that 15 is divisible by 3 and 5, showing that these two numbers are its prime factors.

Prime factorization is vital in various fields, including cryptography, which relies on the difficulty of factoring large composites to keep data secure.

Natural Numbers

Natural numbers are the numbers we naturally count with, starting from 1 and continuing indefinitely—1, 2, 3, and so on. They are the building blocks of our number system and are crucial for understanding prime factorization because prime factorization only applies to natural numbers greater than 1.When working with prime factorization, we only consider the set of positive natural numbers, also called counting numbers, as these are the numbers that can be factored into primes. Natural numbers include both prime and composite numbers, but not fractions, decimals, or negative numbers.

It's important to note that while the number 1 is a natural number, it is neither prime nor composite. It is a unique number with just one positive divisor, which is itself.

Prime Numbers

Prime numbers are the superheroes of the natural numbers world. They are natural numbers greater than 1 that have no divisors other than 1 and themselves. Examples include 2, 3, 5, 7, 11, and so on.

Characteristics of Prime Numbers

Prime numbers are the 'atoms' of the mathematical world. They are indivisible and are the building blocks for all other natural numbers.For example, when you decompose the composite number 15 into its prime factors, you find it is made up of the prime numbers 3 and 5. The prime factorization process is like uncovering the fundamental elements that, when multiplied together, give you the original number. The number 2 is particularly special as it is the only even prime number, all other primes are odd.

In the context of our exercise, recognizing that 15 is made of primes 3 and 5 helps us understand that prime numbers can combine in various ways to form composite numbers, enabling us to delve into more complex mathematical concepts.