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Chapter 1: Problem 18

Label each of the following numbers as prime, composite, or neither. $$ 4 $$

Short Answer

Expert verified
4 is composite.

Step by step solution

01

Understand Definitions

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a natural number greater than 1 that has more than two positive divisors. Numbers that are not considered prime or composite include 0 and 1.
02

Identify the Number

The number to classify is 4.
03

Check for Prime

Determine whether 4 has any divisors other than 1 and 4. If there are no other divisors, it is a prime number. Otherwise, it is not a prime number.
04

List Divisors of 4

Find the divisors of 4. The divisors of 4 are: 1, 2, and 4.
05

Classify the Number

Since 4 has more than two divisors (1, 2, and 4), it is a composite number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of the respective core concept
Let's start by understanding the idea of prime numbers. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In simpler terms, a prime number only has two factors: 1 and itself.
For example:
鈥� 2 is a prime number because its only divisors are 1 and 2.
鈥� 3 is a prime number because its only divisors are 1 and 3.
Notice how there are no other numbers that can divide into these primes without leaving a remainder.
Prime numbers are like the building blocks of natural numbers as every natural number can be factorized into a product of primes.
headline of the respective core concept
Next, let's discuss composite numbers. A composite number is a natural number greater than 1 which is not prime. This means it has more than two positive divisors.
For example:
鈥� 4 is a composite number because its divisors are 1, 2, and 4.
鈥� 6 is a composite number because its divisors are 1, 2, 3, and 6.
Clearly, these numbers can be divided by at least one other number besides 1 and itself.
Composite numbers are important because they can be decomposed into prime factors, revealing the 'ingredients' that make up the number.
headline of the respective core concept
Lastly, let's examine the concept of divisors. Divisors (or factors) are numbers that divide another number completely, leaving a remainder of zero.
For instance, to find the divisors of 4, you need to identify which numbers can be multiplied together to result in 4.
The divisors of 4 are: 1, 2, and 4. This is because:
鈥� 1 * 4 = 4
鈥� 2 * 2 = 4
Divisors help us understand the structure of numbers, as they reveal how numbers can be broken down into simpler multiplication components.
By understanding divisors, you can more easily determine whether a number is prime or composite.

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