Problem 50 - Fundamentals of Mathematics

Question

Determine which of the whole numbers are prime and which are composite. 7

Step-by-Step Solution

Answer

The number 7 is a prime number.

Step 1: Understand What Prime and Composite Numbers Are

A prime number is a number greater than 1 that has no divisors other than 1 and itself. In contrast, a composite number has additional divisors, meaning it can be divided evenly by numbers other than 1 and itself.

Step 2: Examine the Factors of 7

List the numbers that can divide 7 without leaving a remainder. Since 7 can only be divided by 1 and itself (7), it means it's a prime number because it has no other divisors.

Step 3: Conclusion Based on Factors

Since 7 can only be divided by 1 and 7, it fits the definition of a prime number. It has no other divisors.

Key Concepts

Composite Numbers

Understanding the concept of composite numbers is crucial when distinguishing them from prime numbers. Composite numbers are whole numbers that have more than two divisors. This means they have divisors other than just 1 and themselves. For example, the number 4 is a composite number because it can be divided by 1, 2, and 4 without leaving a remainder.

• Composite numbers are always greater than 1.
• They include all non-prime numbers greater than 1.
• Every composite number can be factorized into smaller integers.

Saying a number is composite is essentially stating that it can be built or constructed by multiplying other whole numbers together. A good trick to identify them is to see if you can find any divisors other than 1 and the number itself. If you can, it's composite!

Whole Numbers

Whole numbers are the basic building blocks of arithmetic. They consist of all the natural numbers including zero. This set of numbers starts from 0 and goes on infinitely.

• Whole numbers include: 0, 1, 2, 3, and so on.
• They are non-negative, which means they are either positive or zero.
• Whole numbers are used to count objects or things that can be put into a whole unit.

When working with prime and composite numbers, you'll always be dealing with whole numbers greater than 1. Zero and one are special cases: zero is neither prime nor composite, while one is neither a genuine prime nor composite number as it does not meet the criteria of having exactly two distinct divisors.

Divisors

Divisors play a key role in determining whether a number is prime or composite. A divisor is a number that can divide another number completely, without leaving a remainder. For instance, the divisors of 6 are 1, 2, 3, and 6 since they each can divide 6 without leaving a remainder.

• To find a divisor, see if you can divide two numbers evenly.
• Prime numbers have exactly two divisors: 1 and themselves.
• Composite numbers have more than two divisors.

Learning to identify divisors helps in figuring out whether a number is prime or not. By examining all possible divisors, you can effectively categorize whole numbers into prime or composite. This fundamental concept aids in various areas of mathematics and problem-solving.