Prime Numbers and Prime Factorization

🌟 Definition of Prime numbers:

Prime numbers are natural numbers that have exactly two different factors: 1 and the number itself.

• ✅ 2 is a prime number because it has only two factors: 1 and 2.
• ❌ 4 is not a prime number because it has three factors: 1, 2, and 4.
• ❌ 1 is not a prime number because it only has one factor: 1.

Exam tip: 2 is the smallest and only even prime number.

🧱 Prime numbers are often called the “building blocks” of mathematics because every whole number can be made by multiplying primes.

🌟 Prime Numbers Below 20 Are:

$$2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19$$

🌟 Non-Prime Numbers Below 20 Are:

$$1,\ 4,\ 6,\ 8,\ 9,\ 10,\ 12,\ 14,\ 15,\ 16,\ 18 $$

Exam tip: Try to memorise the prime numbers; they show up a lot in maths exams, especially in questions about factors and multiples.

🌟 What Does Prime Factorization Mean?

Prime factorization means writing a number as a product of prime numbers only.

One method is using division.

🌟 Example of Prime Factorization:

Let’s factorise 90 step by step:

• Start with the smallest prime number (2) that divides 90:

  $$90 = 2 \times 45$$

• Try dividing 45 by the next smallest prime (3):

  $$ 90 = 2 \times 3 \times 15 $$

• 15 is not a prime number, so we keep dividing:

  $$ 90 = 2 \times 3 \times 3 \times 5$$

• All the numbers are now prime. So the final answer in index form is:

$$ 90 = 2 \times 3^2 \times 5 $$

🌟 What Is A Factor Tree?

A factor tree is a visual method to break a number down into its prime factors step by step.

🌟 Example of Prime Factorisation:

Let’s start with 126.

• Start with the smallest prime number (2):
  $$126 = 2 \times 63$$

• Continue factoring the next number (3):
  $$63 = 3 \times 21$$

• 21 is not a prime number, so we continue dividing: 
  $$21 = 3 \times 7$$

• Now, all factors are prime numbers:
  $$126 = 2 \times 3 \times 3 \times 7$$

• The Final answer in index form:
  $$126 = 2 \times 3^2 \times 7$$

Exam tip: This method helps you see the breakdown clearly and is especially useful for math exams.