Problem 23 - Elementary and Intermediate Algebra

Question

Find the prime factorization of each number. If the number is prime, state this. $$ 30 $$

Step-by-Step Solution

Answer

The prime factorization of 30 is 2 × 3 × 5.

Step 1 - Divide by the smallest prime number

Start dividing 30 by the smallest prime number, which is 2. Since 30 is even, it's divisible by 2: 30 ÷ 2 = 15

Step 2 - Continue with the quotient

Now, take the quotient from Step 1, which is 15, and determine if it’s divisible by the smallest prime number. Since 15 is not even, it is not divisible by 2. Move to the next smallest prime number: 3. 15 ÷ 3 = 5

Step 3 - Check the new quotient

Now, take the quotient from Step 2, which is 5, and see if it’s a prime number. 5 is a prime number because it’s only divisible by 1 and itself. So, 30 can be expressed as 30 = 2 × 3 × 5.

Key Concepts

Divisibility

Understanding divisibility is crucial for solving prime factorization problems. Divisibility rules help us determine if one number can be divided by another without leaving a remainder.
For example, the number 30 is divisible by 2, since 30 is even. You can find this by checking if the last digit is one of (0, 2, 4, 6, or 8).
After dividing 30 by 2, we get 15. Now, we check the next smallest prime number (3). Since the sum of 15's digits (1 + 5 = 6) is divisible by 3, 15 is also divisible by 3.
Knowing these rules lets us quickly find the prime factors of a number. You can apply divisibility tests for other numbers like 5, where a number ending in 0 or 5 will be divisible by 5.
Using these rules efficiently simplifies problems and makes factorization easier.

Prime Numbers

Prime numbers are fundamental to factorization. A prime number is a number greater than 1 that has no divisors other than 1 and itself.
Some common prime numbers include 2, 3, 5, 7, 11, 13, etc.
In our example, we used prime numbers 2, 3, and 5 to factorize 30.
Understanding primes is critical because they are the 'building blocks' of all numbers. Multiplying prime numbers in different combinations can construct any given number.
Once you identify a quotient as a prime number, you stop further division. For instance, after dividing 15 by 3 to get 5, we recognize 5 as prime, completing the factorization.
Learning to identify primes quickly aids in efficient solving of mathematical problems involving factorization.

Factorization

Factorization is breaking down a number into its constituent factors. This process can reveal the prime factors of a number.
The prime factorization involves dividing the number by the smallest prime possible and repeating this until you get a prime number.
Let's recall our example: Starting with 30, we divided by 2 to get 15. Then, 15 was divided by 3 to get 5. Since 5 is prime, the factorization of 30 is complete as 2 × 3 × 5.
Factorization: 30 = 2 × 3 × 5
This method can be used for any composite number. If during the process a number turns out to be prime, you know factorization is done.
Factorization is particularly useful in mathematics for simplifying fractions, finding the greatest common divisor, and solving problems involving multiples.