Problem 51 - Intermediate Algebra

Question

The set of prime numbers less than 8

Step-by-Step Solution

Answer

The set of prime numbers less than 8 is \( \{2, 3, 5, 7\} \).

Step 1: Define a Prime Number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Step 2: Identify Natural Numbers Less Than 8

List all natural numbers less than 8: 2, 3, 4, 5, 6, and 7.

Step 3: Check Each Number for Primality

Evaluate each number to see if it is divisible only by 1 and itself: - 2 is only divisible by 1 and 2 - 3 is only divisible by 1 and 3 - 4 is divisible by 1, 2, and 4 (not prime) - 5 is only divisible by 1 and 5 - 6 is divisible by 1, 2, 3, and 6 (not prime) - 7 is only divisible by 1 and 7

Step 4: Compile the Set of Prime Numbers

From the checked numbers, select those that are prime: 2, 3, 5, and 7.

Key Concepts

Natural Numbers

Natural numbers are basically the numbers we count with. Think of them as your starting point when learning to count. These numbers include all positive integers starting from 1 and moving upwards like 1, 2, 3, and so forth. They do not include any fractions, decimals, or negative numbers.
In the context of our problem, when we talk about natural numbers less than 8, we are dealing with the set: 1, 2, 3, 4, 5, 6, and 7. These numbers are essential, as they form the basis of discovering primes and engaging in other mathematical operations. Remember:

• Natural numbers start from 1, not 0.
• No negative numbers are included.
• They are infinite, always going higher with no end.

Divisibility

Divisibility is a key concept when learning about numbers. Simply put, a number is divisible by another if you can divide them evenly, leaving no remainder. This concept helps us determine relationships between numbers.
This is crucial when we evaluate if a number is prime or composite. For instance, consider 6: it is divisible by 1, 2, 3, and 6. When you divide 6 by these numbers, you get whole numbers without fractions, which means that such numbers can divide 6 evenly.
A quick rule of thumb:

• If a number is divisible by any number other than 1 and itself, it is not prime.
• Understanding divisibility helps in simplifying fractions and ratios.
• Use divisibility rules to check numbers quickly, e.g., a number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.

Primality Test

Determining if a number is prime is akin to conducting a primality test. This test involves checking whether a number is divisible only by 1 and itself, which signifies it is a prime.
For numbers less than 8, like 2, 3, 5, and 7, the test is quite straightforward. For instance:

• 2 is only divisible by 1 and 2, making it prime.
• 3 cannot be divided evenly by any number other than 1 and 3, thus it's also prime.

As you conduct these tests, you start by trying to divide with the smallest primes and move upwards if necessary, especially for larger numbers.
Some key points include:

• The smallest prime number is 2, which is also the only even prime.
• All numbers can't be prime; knowing divisibility can quickly rule them out.
• The primality test becomes more complex with larger numbers, often involving special algorithms or sieves like the Sieve of Eratosthenes.

Understanding this concept is not just about numbers but also about the foundational skills in problem-solving and critical thinking in mathematics.