Problem 47 - Fundamentals of Mathematics

Question

Determine which of the whole numbers are prime and which are composite. 2

Step-by-Step Solution

Answer

2 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 has more than two distinct divisors.

Step 2: Check if 2 is Greater than 1

We must first confirm that the number 2 is greater than 1, which it is.

Step 3: Determine the Divisors of 2

List whether 2 can be divided evenly (without a remainder) by any whole numbers other than 1 and 2. For 2, the possible divisors are only 1 and 2.

Step 4: Conclusion on 2's Primality

Since 2 can only be divided evenly by 1 and itself, it meets the definition of a prime number.

Key Concepts

Composite Numbers

Composite numbers are an interesting category of numbers in the world of mathematics. Unlike prime numbers, composite numbers have more than two divisors. This means they can be evenly divided by 1, themselves, and at least one other whole number.

For example, take the number 4. It can be divided by 1, 2, and 4. This array of divisors indicates that 4 is a composite number.

In general, composite numbers are always greater than 1 since 1 itself is neither prime nor composite. Numbers such as 4, 6, 8, 9, and 10 are common examples of composite numbers. They enrich our number system by offering a clear contrast to their prime counterparts.

Divisors

Divisors play a crucial role in identifying whether a number is prime or composite. A divisor is a whole number that can divide another number completely without leaving a remainder.

For instance, if we consider the number 6, its divisors are 1, 2, 3, and 6 because these numbers can divide 6 with no remainder.

To determine the divisors of a number, we can follow these steps:

• Start with the smallest number, 1, and go up to the number itself.
• Check if each of these numbers divides the target number evenly.
• List out all the numbers that divide without a remainder.

The concept of divisors is fundamental when distinguishing between prime and composite numbers.

Whole Numbers

Whole numbers encompass the set of numbers that include zero and all positive integers. They are crucial in the study of basic arithmetic and number theory. Whole numbers can be visualized as points on a number line starting from zero and continuing indefinitely in the positive direction.

Some key characteristics of whole numbers include:

• They do not include fractions or decimals.
• They are non-negative, meaning they start from zero and increase.
• Examples include 0, 1, 2, 3, and so forth.

Whole numbers are vital when we talk about natural numbers like prime and composite numbers since they form the base set from where such numbers are derived. Understanding whole numbers lays the groundwork for grasping more complex mathematical ideas.