Prime Numbers and Prime Factorization

TL;DR

Prime numbers are special whole numbers greater than 1 that only divide evenly by 1 and themselves. Prime factorization breaks any whole number down into a unique set of these prime building blocks. This process is super useful for understanding how numbers relate to each other in topics like HCF and LCM.

1. The Mental Model

Think of prime numbers as the fundamental, unbreakable building blocks of all other whole numbers (except 0 and 1). Prime factorization is like taking apart a LEGO creation to see exactly which basic bricks it's made from.

2. The Core Material

What's a Prime Number?

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A prime number is a whole number greater than 1 that has only two positive divisors (numbers that divide into it evenly): 1 and itself.

• Examples of primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
• Examples of NOT primes:
  • 1 (only one divisor: 1)
  • 4 (divisors: 1, 2, 4)
  • 6 (divisors: 1, 2, 3, 6)

Any whole number greater than 1 that isn't prime is called a composite number. Composite numbers can be broken down into prime factors.

What is Prime Factorization?

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Prime factorization is the process of finding which prime numbers multiply together to make a composite number. Every composite number has one unique prime factorization (ignoring the order of the factors). This is called the Fundamental Theorem of Arithmetic.

You can use a factor tree or repeated division to find prime factors.

Using a Factor Tree

1. Start with the number at the top.
2. Break it down into any two factors.
3. Circle any prime factors you find.
4. Continue breaking down composite factors until all branches end in prime numbers.

Here's how a factor tree works:

graph TD
    A["Number to Factor"] --> B["Factor 1 (prime?)"]
    A --> C["Factor 2 (prime?)"]
    B --> D{"Prime Factor!"}
    C --> E["Factor 3 (prime?)"]
    C --> F["Factor 4 (prime?)"]
    E --> G{"Prime Factor!"}
    F --> H{"Prime Factor!"}

Using Repeated Division

1. Start with the number.
2. Divide it by the smallest prime number that divides into it evenly.
3. Write down the prime factor and the result.
4. Repeat the process with the result until you can't divide anymore.

Example: Prime factorize 60.

• 60 ÷ 2 = 30
• 30 ÷ 2 = 15
• 15 ÷ 3 = 5
• 5 ÷ 5 = 1

So, the prime factors are 2, 2, 3, and 5.

Writing Prime Factorization

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Once you have the prime factors, you write them as a product, often using exponents for repeated factors.

For 60: $2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$.

3. Worked Example

Let's find the prime factorization of 84 using a factor tree.

1. Start with 84.
2. 84 can be split into 2 and 42. (2 is prime, so circle it).
3. 42 can be split into 2 and 21. (2 is prime, circle it).
4. 21 can be split into 3 and 7. (Both 3 and 7 are prime, circle them).

The prime factors are 2, 2, 3, and 7.
So, the prime factorization of 84 is $2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7$.

4. Key Takeaways

• A prime number is a whole number greater than 1 with only two divisors: 1 and itself.
• 1 is not a prime number; 2 is the smallest and only even prime number.
• Prime factorization breaks down a composite number into a unique set of prime numbers multiplied together.
• You can use factor trees or repeated division to find the prime factors of a number.
• Always write prime factorizations using exponents for repeated prime factors.
• Prime factorization is a foundational skill for understanding HCF and LCM.
• Every whole number greater than 1 is either prime or can be written as a unique product of primes.

Common Mistakes to Avoid:
- Don't include 1 in your list of prime factors; it's not prime.
- Make sure all your final factors are actually prime numbers.
- Forgetting to use exponents for repeated prime factors (e.g., writing $2 \times 2 \times 3$ instead of $2^2 \times 3$).
- Only using even numbers for division; remember to try 3, 5, 7, etc.

5. Now Try It

Find the prime factorization of the number 120. Use either a factor tree or repeated division, and then write your final answer using exponents.

What success looks like: You'll have a product of prime numbers (like $2^a \times 3^b \times 5^c \ldots$) that, when multiplied out, equals 120. Make sure all the bases are prime numbers and the exponents are correct.