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:
- Start with the smallest prime number, which is 2.
- Divide the original number by this prime number.
- If the number is divisible, write down the prime number and divide the original number by this prime.
- Repeat the process with the quotient until it is not divisible by 2.
- Move on to the next smallest prime number (3, 5, 7, 11, …) and repeat the steps.
- 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.
- Start with 2. Is 60 divisible by 2? Yes, it is. \( 60 ÷ 2 = 30 \)
- Now, take 30. Is it divisible by 2? Yes! \( 30 ÷ 2 = 15 \)
- 15 is not divisible by 2, so let’s move to the next prime number, which is 3. \( 15 ÷ 3 = 5 \)
- 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:
- Find the prime factorization of 28.
- What is the prime factorization of 45?
- Determine the prime factors of 100.
Solutions:
- 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 \( 2^2 × 7 \).
- 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 \( 3^2 × 5 \).
- 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!
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