Problem 20 - 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. $$100$$

Step-by-Step Solution

Answer

The number 100 is a composite number. Its prime factorization is \(2^{2}*5^{2}\).

Step 1: Identify Number Type

Firstly, let's identify 100 as a prime or composite number. The number 100 can be divided evenly by 1, 2, 4, 5, 10, 20, 25, 50 and 100, with no remainder. Since 100 has more than two different factors, it is a composite number.

Step 2: Prime Factorization

The next step is to perform the prime factorization for 100. The process involves dividing 100 by prime numbers starting from the smallest prime number, which is 2. Continue this process until the result is a prime number. In this case, when we perform this process, we get that 100 = 2 * 2 * 5 * 5.

Step 3: Write Down the result

So, we have that the prime factorization of 100 is \(2^{2}*5^{2}\). Hence, it indicates that the number 100 is not a prime number but a composite number with its prime factors being 2 and 5.

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime factors. A prime number is a number greater than 1 that has no divisors other than 1 and itself. To start with prime factorization, you take a composite number, such as 100, and break it down into the smallest set of prime numbers that will multiply together to give the original number. This helps in simplifying numbers or solving mathematical problems that involve factors.

To factorize 100, we begin by dividing it by the smallest prime number, which is 2. Since 100 is even, it's divisible by 2. We get 50, then divide 50 by 2 to get 25. Next, 25 is divided by the next prime number, which is 5, resulting in 5. Since 5 is itself a prime number, we stop there. Thus, we express 100 as the prime factorization equation: \(100 = 2^2 \times 5^2\).

In the equation, the exponents tell us how many times each prime number is used as a factor. It's a neat and organized way to break down composite numbers into their prime components.

Natural Numbers

Natural numbers refer to the set of positive integers starting from 1 upwards, like 1, 2, 3, and so on. They are foundational in mathematics for counting and ordering. Natural numbers are infinite, meaning they continue indefinitely without end.

Importantly, in the context of this exercise, we deal with natural numbers when identifying if a given number is prime or composite. To determine the type of number, you check its divisors. For example, with the natural number 100, we realize it's composite because it has several divisors other than 1 and itself — including 2, 4, 5, 10, 25, and 50.

Natural numbers are considered the simplest form of numbers and are particularly handy because they do not include fractions or decimals. They form the basis for other number sets, such as whole numbers and integers.

Prime Numbers

Prime numbers are natural numbers greater than 1, characterized by having exactly two distinct positive divisors: 1 and the number itself. This means a prime number cannot be divided evenly by any other natural numbers. Examples include 2, 3, 5, 7, and 11.

In mathematical problems, prime numbers are vital for various functions like cryptography, number theory, and algebraic structures. They are often used in prime factorization. As we see in the step-by-step solution of the exercise with the number 100, prime numbers play a significant role in breaking down composite numbers.

While 100 isn't a prime number, its prime factors — 2 and 5 — are essential in understanding its structure. Recognizing prime numbers helps in determining the prime factorization of any composite number, helping us solve mathematical puzzles and problems more effectively. This highlights their fundamental importance in math.