Problem 2 - 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. $$23$$

Step-by-Step Solution

Answer

23 is a prime number as it has no divisors other than 1 and 23.

Step 1: Understand Definitions

A prime number is a whole number greater than 1, which has no divisors other than 1 and itself. A composite number is a whole number greater than 1 that has more than two divisors.

Step 2: Check for Divisibility

To determine if 23 is a prime number, check whether it is divisible by any numbers other than 1 and 23. The most efficient way to check prime status is to test divisibility with prime numbers up to the square root of the number.

Step 3: Test Smallest Prime Numbers

Test the divisibility of 23 by the smallest prime numbers, which are 2, 3, and 5. Since 23 is odd, it is not divisible by 2. Next, add the digits of 23 (2 + 3 = 5), which is not divisible by 3, so 23 is not divisible by 3 either. 23 does not end in 0 or 5, so it is not divisible by 5.

Step 4: Conclusion

Since 23 is not divisible by the prime numbers 2, 3, or 5, 23 has no divisors other than 1 and itself. Therefore, 23 is a prime number.

Key Concepts

Composite Numbers

Composite numbers are fascinating because they are the opposite of prime numbers. Unlike prime numbers, composite numbers have more than two factors. That means the number can be divided by one, itself, and at least one other number.
For any whole number greater than 1, if there’s an ability to find a divisor other than 1 and the number itself, it’s called a composite number.
Here are a few key points to remember about composite numbers:

• Examples of composite numbers include 4, 6, 8, 9, 10, and 12.
• Each of these numbers can be divided by numbers other than 1 and itself. For instance, 6 can be divided by 2 and 3.
• The number 4, for example, has three divisors: 1, 2, and 4.

Composite numbers become easier to understand once you realize they are all about having multiple factors.

Divisibility Rules

Divisibility rules are handy shortcuts that help you determine whether one number is divisible by another without performing complex division. This is particularly useful for figuring out if a number is prime or composite. Each number has its own set of rules:

• Divisibility by 2: If the last digit is even, then the number is divisible by 2.
• Divisibility by 3: If the sum of the digits is divisible by 3, then the original number is too.
• Divisibility by 5: If the last digit is 0 or 5, the number is divisible by 5.
• Testing other numbers: For numbers like 23, check divisibility using prime numbers below its square root.

Using these rules makes the process faster and less cumbersome, especially when determining if a number might be composite.

Factors

Factors are numbers that divide evenly into another number without leaving a remainder. They are an essential part of understanding both prime and composite numbers.
Here is what you need to know about factors:

• For any given number, like 10, factors include numbers such as 1, 2, 5, and 10.
• Every number has at least two factors: 1 and the number itself.
• A number's factors can be found by checking which numbers divide evenly into it.

When you list out factors, it helps to reveal whether a number is prime or composite. A prime number will only have two factors, whereas a composite will have more.