Problem 25 - 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. $$81$$

Step-by-Step Solution

Answer

The number 81 is composite. Its prime factorization is \( 3^4 \).

Step 1: Identification: Prime or Composite

Begin by checking if 81 has any divisors between 2 and \( \sqrt{81} \). The square root of 81 is 9. Check each whole number from 2-9. It is found that 3, 9, and 27 divide 81, therefore, it is a composite number.

Step 2: Prime Factorization

To find the prime factorization of 81, start by dividing it by the smallest prime number, which is 2. If it is not divisible by 2, move to the next prime numbers, which are 3, 5, 7, and so on. Continuing this process, the prime factorization of 81 is found to be \( 3 \times 3 \times 3 \times 3 \) or \( 3^4 \).

Key Concepts

Composite Numbers

When discussing composite numbers, it is important to understand that these numbers are natural numbers with more than two distinct positive divisors. In simple terms, a composite number is a whole number that can be divided evenly by numbers other than 1 and itself. Composite numbers are the opposite of prime numbers, which have exactly two distinct positive divisors.

For example, the number 81 is a composite number. When you try dividing it by numbers other than 1 and 81 itself, you find that it can be divided evenly by 3, 9, and 27. This means that 81 has multiple divisors, confirming its status as a composite number. Unlike prime numbers, composites are more commonly found as we move beyond the number 1.

To determine if a number like 81 is composite, you check for divisibility by numbers up to its square root. If you find any divisors, it is composite. This process helps identify composite numbers efficiently.

Natural Numbers

Natural numbers are the simplest form of numbers and are part of a system of numbers used for counting and ordering. They begin from 1 and extend infinitely in a positive direction. Natural numbers include all the positive integers, and they are the foundation of most numeric concepts used in mathematics.

These numbers are used every day for basic counting, like tallying items or ranking order. For example, 1, 2, 3, 4, and so on are all natural numbers. The number 81 is also a natural number because it follows the basic rule of being a positive integer.

Natural numbers have no fractional or decimal component. They are whole numbers, making them easy to understand and use in practical situations. The concept of natural numbers is critical when exploring more complex mathematical ideas, including composite numbers and factorization.

Prime Numbers

Prime numbers are key elements in the study of mathematics, especially because of their role in number theory. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, a prime number only has two distinct positive divisors: 1 and itself.

Some of the smallest prime numbers include 2, 3, 5, 7, and 11. For example, the number 3 is prime because it cannot be divided by any number other than 1 and 3 without leaving a remainder.

Prime numbers are foundational because they are essentially the "building blocks" of other numbers. Every composite number, such as 81, can be represented as a product of prime numbers. This factorization process, like what was done with 81 to get \( 3^4 \), is crucial in various mathematical applications, including cryptography and algorithms.