Problem 5 - Prealgebra

Question

Identify each of the numbers below as either a prime number or a composite number. For those that are composite, give at least one divisor (factor) other than the number itself or the number 1. $$81$$

Step-by-Step Solution

Answer

81 is a composite number with 3 as a divisor.

Step 1: Understanding Prime and Composite Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a natural number that has divisors other than 1 and itself.

Step 2: Check if 81 is Prime or Composite

To determine if 81 is prime, we need to see if there are any divisors other than 1 and 81. We'll start by testing small prime numbers like 2, 3, and 5.

Step 3: Division by 2

81 is odd, so it is not divisible by 2.

Step 4: Division by 3

To see if 81 is divisible by 3, add the digits: 8 + 1 = 9. Since 9 is divisible by 3, 81 is divisible by 3.

Step 5: Divide 81 by 3

Divide 81 by 3 to find a factor: \[81 \div 3 = 27\]. Since 27 is not 1 or 81, 3 is a factor, confirming 81 is composite.

Key Concepts

Prime Numbers

Prime numbers are fascinating creatures in the realm of mathematics. These numbers are only divisible by 1 and themselves. This means, there are no other natural numbers that can evenly divide them without leaving a remainder.
For instance, if you take the number 5, the only numbers that can divide it without leaving a remainder are 1 and 5 itself. This makes 5 a prime number.

• They must be greater than 1.
• Cannot be divided evenly by other numbers except for 1 and themselves.

The first few prime numbers are 2, 3, 5, 7, and 11. Notice how 2 is unique—it’s the only even prime number. All other even numbers can be divided by 2, which makes them composite. Knowing the concept of prime numbers is essential in different areas of math, especially in understanding their role in number theory and cryptography.

Composite Numbers

Composite numbers are the friendly counterparts of prime numbers. Unlike primes, composites have more than two factors. In other words, they can be divided evenly by numbers other than 1 and themselves.
For example, 8 is a composite number because it can be divided evenly by 1, 2, 4, and 8. The key property of composite numbers is that they have additional divisors, making them non-prime.

• They have more than two factors.
• Always have at least one divisor other than 1 and itself.

Currently, the smallest composite number is 4, as it is divisible by 1, 2, and 4. Understanding composite numbers helps in various calculations and problem-solving scenarios, including factorization used in higher mathematics.

Factors and Divisors

Understanding the terms "factors" and "divisors" is crucial when discussing prime and composite numbers. These terms help us break down numbers into their building blocks.
Factors are the numbers you multiply together to get another number. For example, the factors of 18 are 1, 2, 3, 6, 9, and 18 because 2 times 9 equals 18 and 3 times 6 equals 18. Divisors are similar—they refer to any number that divides another number completely without leaving a remainder.

• Every number has at least two factors: 1 and itself.
• Divisors come in pairs; if 3 is a divisor of 12, so is 4 since 3 times 4 equals 12.

Being able to determine factors and divisors is integral for identifying prime and composite numbers and solving equations that require factorization and simplification. It empowers students to excel in mathematical computations and problem-solving.