Question

Write three descriptions of the elements of the set 12,5,8,11,14\(\\}\)

Short answer

The set contains even numbers (12, 8, 14) and prime numbers (5, 11).

Step-by-step solution

Identify each element

The given set contains five elements: 12, 5, 8, 11, and 14. Let's take a look at each entry individually in the set.

Describe Element 1

The first element is 12. It is an even number and can be expressed as a product of its prime factors: \( 12 = 2^2 \times 3 \).

Describe Element 2

The second element is 5. It is a prime number, which means its only divisors are 1 and itself.

Describe Element 3

The third element is 8. It is an even number and can be expressed as a power of 2: \( 8 = 2^3 \).

Describe Element 4

The fourth element is 11. It is a prime number, meaning it has no divisors other than 1 and itself.

Describe Element 5

The fifth element is 14. It is an even number and can be factored into primes as: \( 14 = 2 \times 7 \).

Key concepts

Prime Numbers

Prime numbers are the building blocks of all numbers. They are defined as numbers greater than 1 that have no divisors other than 1 and themselves. This uniqueness means they cannot be formed by multiplying two smaller natural numbers.
Prime numbers are fundamental in mathematics because they are the "atoms" from which all other numbers (called composite numbers) are made. The number 2 holds a special place as the only even prime number.
Several key properties help to identify prime numbers:

• They must be greater than 1.
• They cannot be divided evenly by any integer other than 1 and themselves.
• When checking if a number is prime, you only need to test divisibility by prime numbers up to the square root of the number in question.

A couple examples of prime numbers include 5 and 11. In prime factorization, these prime numbers serve as the smallest factors that compose larger numbers. Understanding what makes a number prime is crucial for various areas of mathematics, both theoretical and practical.

Prime Factorization

Prime factorization is the process of breaking down a composite number into a product of prime numbers. This method helps in simplifying mathematical expressions and solving equations where multiplication or division of large numbers is involved.
To perform prime factorization:

• Begin with the smallest prime number, 2, and check if it divides the given number.
• If it does, then divide the number by 2, and repeat the process with the resulting quotient until 2 no longer divides evenly.
• Move to the next smallest prime number, such as 3, and continue the process.
• Keep dividing by increasingly larger prime numbers, until the quotient becomes 1.

For example, the number 12 can be broken down into its prime factors as: 12 = 2 × 2 × 3 or 12 = 2^2 × 3. This factorization represents 12's simplest prime form. Similarly, 14 can be broken down into primes as 14 = 2 × 7. Understanding prime factorization is not only about writing numbers differently, but it is also about grasping why some numbers divide others and finding least common multiples or greatest common divisors in number theory and algebra.

Even Numbers

Even numbers are integers that are perfectly divisible by 2. This feature makes them easily recognizable. Any number that can be divided by 2 without leaving a remainder is considered even.
Some properties of even numbers are:

• The number 2 is the first and smallest even number, and perhaps the most significant because it is also the only even prime number.
• Even numbers occur every other number in the number line.
• When two even numbers are added or subtracted, the result is always an even number again.
• Multiplying any number by 2 always results in an even number.

Examples from the given set include the numbers 12, 8, and 14. Each can be factored into smaller primes where the factor 2 is present, showcasing their even nature: 8 = 2^3, 12 = 2^2 × 3, and 14 = 2 × 7. Even numbers form an essential part of mathematics, often making calculations easier due to their regularity and predictability in division and multiplication operations.