Problem 8 - Fundamentals of Mathematics

Question

Determine which of the following whole numbers are prime and which are composite. 3

Step-by-Step Solution

Answer

3 is a prime number.

Step 1: Understand Prime and Composite Numbers

A prime number is a whole number greater than 1 that has no divisors other than 1 and itself. A composite number is a whole number greater than 1 that is not prime; it has divisors other than 1 and itself.

Step 2: Determine Divisors of 3

Identify the divisors of the number 3. Check if 3 has any divisors other than 1 and 3. Since 3 can only be divided by 1 and 3 itself without leaving a remainder, list these divisors.

Step 3: Classify the Number as Prime or Composite

Since the only divisors of 3 are 1 and 3, 3 cannot be divided exactly by any other whole number. Thus, 3 is a prime number.

Key Concepts

Composite Numbers

Composite numbers are the opposite of prime numbers in a sense. They are whole numbers greater than 1 that can be divided by numbers other than just 1 and themselves. This means composite numbers have multiple divisors. For example, while a prime number like 3 only has two divisors (1 and 3), a simple composite number like 4 can be divided by 1, 2, and 4.
This characteristic of having additional divisors distinguishes composites and is the root of their mathematical significance. Composite numbers start from 4, after all possible attempts to divide a number by smaller numbers result in an integer example: 4, 6, 8, 9, and so on.
Remember that the smallest composite number is 4, and any whole number greater than 1 that isn't a prime will be composite.
Below are a few examples to guide you in identifying composite numbers:

• 4: Divisible by 1, 2, and 4
• 6: Divisible by 1, 2, 3, and 6
• 9: Divisible by 1, 3, and 9

Understanding these properties can help when working with numbers and verifying if a number fits the composite category.

Divisors

Divisors are numbers that divide another number completely, which means dividing without leaving a remainder. For any whole number, there are typically multiple divisors, especially if it is a composite number. To find divisors, start by checking smaller numbers to see if dividing them gives an integer.
For instance, when considering the number 10, its divisors include 1, 2, 5, and 10, since these numbers divide 10 without a remainder:

• 10 ÷ 1 = 10
• 10 ÷ 2 = 5
• 10 ÷ 5 = 2
• 10 ÷ 10 = 1

For a prime number, the list of divisors is very short—only 1 and the number itself. Similarly, when determining if a number is composite or prime, observing the amount and nature of its divisors provides a clear answer. Thus, examining a number's divisors is a key step in understanding its mathematical properties.

Whole Numbers

Whole numbers form the basic building blocks of arithmetic. They consist of 0 and all the positive numbers that follow, like 1, 2, 3, and so on, excluding fractions and decimals.
Importantly, when discussing prime and composite numbers, only whole numbers greater than 1 are considered. The concept of divisors is particularly tied to whole numbers.
Imagine whole numbers as steps or units that are applied in counting and calculations.

• Examples include 0, 1, 2, 3, 4, 5, and so forth.
• They may be positive or zero, but never negative.
• Whole numbers are used in every foundational mathematical operation.

Understanding whole numbers is crucial because they serve as the template for discussing more complex mathematics, especially in elementary and middle school education.