Problem 7 - 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. $$13$$

Step-by-Step Solution

Answer

13 is a prime number.

Step 1: Understand Prime and Composite Numbers

Prime numbers are natural numbers greater than 1, which are divisible only by 1 and themselves. Composite numbers have more than two factors—1, the number itself, and at least another proper divisor.

Step 2: Test Divisibility

To determine whether 13 is a prime or composite number, we need to see if it can be divided evenly by any numbers other than 1 and itself. Start testing from the smallest prime number, which is 2, and go up to the square root of 13.

Step 3: Check Divisibility by 2

Since 13 is an odd number, it is not divisible by 2.

Step 4: Check Divisibility by 3

Sum the digits of 13, which is 1 + 3 = 4. Since 4 is not divisible by 3, 13 is not divisible by 3.

Step 5: Conclude the Nature of the Number

Since 13 is not divisible by any of the numbers less than its square root other than 1, it is only divisible by 1 and itself. Therefore, 13 is a prime number.

Key Concepts

Composite Numbers

Composite numbers are fascinating because they demonstrate more complexity compared to prime numbers. A composite number is a natural number greater than 1 that can be divided by numbers other than 1 and itself.
To figure out if any number is composite, check if it has any factors apart from 1 and itself. These factors can be any other natural numbers.

• If a number has more than two factors, it's composite.
• For example, 4 is a composite number since it can be divided by 1, 2, and 4.
• Another example is 12, which has divisors such as 1, 2, 3, 4, 6, and 12.

Identifying composite numbers can sometimes be easier than spotting primes because once you find one additional factor, you know it's not prime.

Divisibility Rules

Understanding divisibility rules can save you a lot of time when determining if a number is prime or composite. These rules allow you to quickly identify whether a number can be divided evenly by another without diving head first into division.

• Divisibility by 2: A number is divisible by 2 if it is even. Therefore, any number ending in 0, 2, 4, 6, or 8 is divisible by 2.
• Divisibility by 3: For a number to be divisible by 3, the sum of its digits must be divisible by 3. For example, consider 111. The sum of its digits is 1 + 1 + 1 = 3, so 111 is divisible by 3.
• Divisibility by 5: Numbers that end in 0 or 5 are divisible by 5, such as 20, 25, or 50.

These rules are excellent shortcuts, particularly when dealing with longer or more challenging numbers. Remember that a prime number won’t satisfy these rules for any factor other than 1 and itself.

Natural Numbers

Natural numbers are the simplest and most innate numbers you'll encounter in mathematics. They start from 1 and go on indefinitely (1, 2, 3, 4, ...). These numbers are used in counting and ordering.
Natural numbers form the basis of most mathematical functions and are separate from negative numbers, fractions, or decimals.
Each natural number can either be a prime number or a composite number greater than 1.

• A natural number greater than 1 is either prime, having no divisors other than 1 and itself, or composite, having additional divisors.
• The number 1, however, is unique as it is neither prime nor composite.

This simple system of natural numbers provides a strong foundation for understanding more complex concepts like integers, rational numbers, and real numbers.