Problem 58 - Algebra 1: Concepts and Skills

Question

Determine whether the number is prime or composite. If it is composite, give its prime factorization. $$ 28 $$

Step-by-Step Solution

Answer

The number 28 is a composite number and its prime factorization is \(2^2 \times 7\).

Step 1: Determine if the number is prime

To check if a number is prime, it should be divided by numbers from 2 to its square root. If at any point it gets divided evenly (i.e., without a remainder), it is not prime. For 28, the square root is approximately 5.29. Hence, need to try dividing 28 by numbers from 2 to 5. Dividing 28 by 2 leaves no remainder, hence 28 is not a prime number.

Step 2: Find the prime factorization of the number

Since 28 is not a prime number, its prime factorization should be found. Start by dividing by the smallest prime number i.e., 2. So, 28 ÷ 2 = 14. Now repeat the process with 14. We get 14 ÷ 2 = 7. Now, 7 is a prime number, so the division process stops here. Hence, the prime factorization of 28 is \(2^2 \times 7\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into a product of prime numbers. It's like finding the building blocks of a number. For the number 28, the prime factorization process begins by identifying the smallest prime number that divides evenly into 28 without leaving a remainder. In this case, we start with 2.

• Divide 28 by 2 to get 14.
• Repeat, dividing 14 by 2, resulting in 7.
• Since 7 is a prime number, the process stops.

Thus, the prime factorization of 28 is expressed as \(2^2 \times 7\). This means that if you multiply 2 twice and then by 7, you will obtain the original number, 28. Prime factorization helps us understand and express numbers by their essential prime components.

Prime Numbers

Prime numbers are special numbers greater than 1, which have no divisors other than 1 and themselves. They retain uniqueness because they cannot be formed by multiplying other numbers together. A prime number, say 7, cannot be divided by any number other than 1 and 7 without leaving a remainder. Prime numbers are the essential building blocks in mathematics. That's why they appear in areas like cryptography and number theory.

• Prime numbers include numbers like 2, 3, 5, 7, and so on.
• 2 is the only even prime number.
• All other even numbers can be divided by 2, making them composite.

These numbers make up the foundation for topics such as prime factorization, where we decompose numbers into their prime parts.

Composite Numbers

Composite numbers are the opposite of prime numbers, as they can be divided by numbers other than 1 and themselves. This means they are made up of two or more prime numbers, making them perfect candidates for prime factorization. For the number 28, we see that it can be divided by numbers other than 1 and 28, like 2 and 7.

• 28 can be factored into prime numbers: \(2^2\) and 7.
• This shows that its prime factorization is \(2^2 \times 7\).
• Every composite number has a unique prime factorization.

Because composite numbers can be factored down to prime elements, they play a crucial role in mathematical calculations and problem solving, providing a structured understanding of how numbers are constructed.