Problem 19 - Elementary and Intermediate Algebra

Question

Label each of the following numbers as prime, composite, or neither. $$ 40 $$

Step-by-Step Solution

Answer

40 is a composite number.

Step 1: Understand the 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 prime or composite are 0 and 1.

Step 2: Check for Special Cases

Confirm that the given number (in this case, 40) is greater than 1. This ensures it's not one of the special cases (0 or 1) which are neither prime nor composite.

Step 3: Determine if 40 is Prime

To determine if 40 is a prime number, check if it has any divisors other than 1 and itself. Start by checking smaller prime numbers (2, 3, 5, etc.) to see if 40 is divisible by any.

Step 4: Divisibility Test

40 is divisible by 2 because 40 ÷ 2 = 20, which means 40 has divisors other than 1 and itself. Thus, 40 is not a prime number.

Step 5: Classification as Composite Number

Since 40 has divisors other than 1 and itself, it meets the criteria for a composite number. Therefore, 40 is a composite number.

Key Concepts

prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means that a prime number cannot be divided evenly by any other number except for 1 and the number itself. For example, 2, 3, 5, 7, and 11 are prime numbers.
To identify a prime number, you can use the divisibility test by checking small prime numbers like 2, 3, 5, and so on. If none of these smaller prime numbers divide the number evenly, then it is prime.

composite number

A composite number is a natural number greater than 1 that has more than two positive divisors. This means that a composite number can be divided evenly by numbers other than 1 and itself.
For instance, 4 is a composite number because it can be divided by 1, 2, and 4. Other examples of composite numbers include 6, 8, 9, and 10.
Composite numbers are essentially the opposite of prime numbers, as they have multiple factors.

divisibility test

The divisibility test is a tool to determine if a number can be divided evenly by another number. This is an essential method for identifying whether a given number is prime or composite.
To perform a divisibility test, you need to check if the number can be divided without a remainder. For example, to see if 40 is divisible by 2, you perform the division: 40 ÷ 2 = 20, which is an integer value with no remainder. Hence, 40 is divisible by 2.
Similarly, you can check divisibility by other small prime numbers like 3, 5, etc. If the number passes the divisibility test with any of these primes, it is composite.

number classification

Number classification involves determining whether a number is prime, composite, or neither.
Generally, natural numbers greater than 1 are either prime or composite. Prime numbers have exactly two divisors (1 and itself), and composite numbers have more than two divisors.
The numbers 0 and 1 are neither prime nor composite. They are special cases in number classification.
To classify a given number, you first check if it is greater than 1, then determine if it has any divisors other than 1 and itself. If it does, it's composite.