Problem 30 - Elementary and Intermediate Algebra

Question

Find the prime factorization of each number. If the number is prime, state this. $$ 120 $$

Step-by-Step Solution

Answer

120 = 2^{3} × 3 × 5

Step 1: Find the smallest prime number

Start by finding the smallest prime number that divides 120 without leaving a remainder. The smallest prime number is 2.

Step 2: Divide by the smallest prime

Divide 120 by 2: ewline 120 ÷ 2 = 60

Step 3: Repeat the process

Continue dividing by 2 until it no longer divides evenly: ewline 60 ÷ 2 = 30 ewline 30 ÷ 2 = 15

Step 4: Move to the next smallest prime

Now, 15 is no longer evenly divisible by 2. Move to the next smallest prime, which is 3, and divide: ewline 15 ÷ 3 = 5

Step 5: Check if the resulting number is prime

Finally, 5 is a prime number. Therefore, the prime factorization of 120 is complete.

Step 6: Write the prime factorization

The prime factorization of 120 can be written as: ewline ewline 120 = 2^{3} × 3 × 5

Key Concepts

Divisibility

Divisibility is an essential concept in mathematics, especially when working with prime factorization. A number is divisible by another if, after dividing them, there is no remainder. For example, 120 is divisible by 2 because when you divide 120 by 2, the result is 60 with no remainder. Divisibility helps determine which smaller numbers (factors) can evenly divide a larger number. This concept is used repetitively until the largest number or factor cannot be divided further. Practicing dividing by smaller prime numbers will help you understand this easily.

Prime Numbers

Prime numbers are the building blocks of all numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. For example, 2, 3, 5, 7, etc., are prime numbers. The number 2 is the smallest and the only even prime number. While factoring, always start with the smallest prime number, 2, then move to the next smallest primes like 3, 5, 7, and so on, until you can’t divide any further. Understanding prime numbers is essential because they simplify the process of breaking down a larger number into its base components.

Factors

Factors are the numbers you multiply together to get another number. For instance, the factors of 120 include 2, 3, and 5. When you multiply these prime factors, you get back the original number: 2 × 2 × 2 × 3 × 5 = 120. Finding factors involves dividing the original number by smaller prime numbers repeatedly until you can’t divide anymore. Factors help simplify complex arithmetic by breaking down larger numbers into manageable pieces and understanding the core prime components of any composite number.

Mathematical Steps

Understanding the step-by-step method is crucial when finding the prime factorization of a number like 120. Here is a simplified explanation of the original steps:

1. **Divide by 2:** 120 ÷ 2 = 60

2. **Divide 60 by 2:** 60 ÷ 2 = 30

3. **Divide 30 by 2:** 30 ÷ 2 = 15 Notice you continue with 2 until it doesn't divide evenly.

4. **Move to the next smallest prime, 3:** 15 ÷ 3 = 5

5. **Check if the resulting number, 5, is a prime:** Yes, since 5 only has divisors 1 and 5.

Finally, you collect all the prime factors to write: 120 = 2^{3} × 3 × 5 Breaking down each step reinforces your understanding of both the process and the underlying principles.