Prime Factorization

---

Understanding Prime Factorization

Today we are going to explore an exciting concept in mathematics: Prime Factorization. This is a way to break down a number into its most basic building blocks – prime numbers. Prime numbers are like the atoms of mathematics; they are numbers that are greater than 1 and can only be divided evenly by 1 and themselves.

What is Prime Factorization?

Prime factorization is the process of finding which prime numbers multiply together to make the original number. It’s like finding the recipe for a number! For example, when we bake a cake, we combine ingredients like flour, sugar, and eggs. Similarly, with prime factorization, we combine prime numbers to create our original number.

Steps for Prime Factorization:

1. Start with the smallest prime number, which is 2.
2. Divide the original number by this prime number.
3. If the number is divisible, write down the prime number and divide the original number by this prime.
4. Repeat the process with the quotient until it is not divisible by 2.
5. Move on to the next smallest prime number (3, 5, 7, 11, …) and repeat the steps.
6. Continue until the quotient is a prime number.

Once you cannot divide any further, you have completed the prime factorization.

Example:

Let’s find the prime factorization of 60.

1. Start with 2. Is 60 divisible by 2? Yes, it is. \( 60 ÷ 2 = 30 \)
2. Now, take 30. Is it divisible by 2? Yes! \( 30 ÷ 2 = 15 \)
3. 15 is not divisible by 2, so let’s move to the next prime number, which is 3. \( 15 ÷ 3 = 5 \)
4. Now, 5 is a prime number itself, so we stop here.

Our prime factors are 2, 2, 3, and 5. So, we write the prime factorization of 60 as \( 2^2 × 3 × 5 \).

Possible Questions from Learners:

Q: What if the number is a prime number itself?

A: If the number is a prime number, then its prime factorization is just the number itself because it cannot be divided into smaller prime numbers.

Q: Do we always start with 2?

A: Yes, we start with 2 because it’s the smallest prime number, and it’s the only even prime number. It simplifies the process before we move on to the odd prime numbers.

Q: How do we know when we’ve finished the prime factorization?

A: We’ve finished when the final quotient is a prime number. At that point, it can no longer be divided by any prime numbers except itself.

Practice Problems:

Now it’s your turn to try some prime factorization. Here are a few problems to solve:

1. Find the prime factorization of 28.
2. What is the prime factorization of 45?
3. Determine the prime factors of 100.

Solutions:

1. Prime factorization of 28:28 is divisible by 2 (28 ÷ 2 = 14), 14 is divisible by 2 (14 ÷ 2 = 7), and 7 is a prime number. So, the prime factorization is \( 2^2 × 7 \).
2. Prime factorization of 45:45 is divisible by 3 (45 ÷ 3 = 15), 15 is divisible by 3 (15 ÷ 3 = 5), and 5 is a prime number. So, the prime factorization is \( 3^2 × 5 \).
3. Prime factorization of 100:100 is divisible by 2 (100 ÷ 2 = 50), 50 is divisible by 2 (50 ÷ 2 = 25), 25 is divisible by 5 (25 ÷ 5 = 5), and 5 is a prime number. So, the prime factorization is \( 2^2 × 5^2 \).

Great job today, class! Remember, practice makes perfect. Keep factoring!

---

The Most Popular Topics:

---

Return from Prime Factorization to MathQuadrum.com home for more topics on mathematics