Prime Numbers and Prime Factorization

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.

Here Are Prime Numbers Below 20:

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

Here Are Non-Prime Numbers Below 20:

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

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

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

One method is using division.

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:

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

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

Example: 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.