Prime factorisation

Key Notes:

What is Prime Factorisation?

Prime Factorisation is the process of breaking down a number into the set of prime numbers that, when multiplied together, give the original number.

Key Concepts
Prime Numbers:
  • Definition: Numbers greater than 1 that have exactly two factors: 1 and themselves.
  • Examples: 2, 3, 5, 7, 11, 13, 17, etc.
Composite Numbers:
  • Definition: Numbers that have more than two factors.
  • Examples: 4, 6, 8, 9, 10, 12, etc.
Factors:
  • Definition: Numbers that divide another number exactly (without leaving a remainder).
  • Example: Factors of 12 are 1, 2, 3, 4, 6, 12.
How to Find Prime Factorisation

Start with the Number:

  • Choose a number you want to factorise.

Divide by the Smallest Prime Number:

  • Divide the number by the smallest prime number (2) if possible.
  • If not, move to the next smallest prime number (3), and so on.

Continue Until the Quotient is a Prime Number:

  • Keep dividing by the smallest prime number until you are left with 1.

Write Down the Prime Factors:

  • List all the prime numbers you used to divide the original number.

Express as a Product:

  • Write the number as the product of its prime factors.
Example:

Find the Prime Factorisation of 36

  • Start with 36.
  • Divide by 2 (the smallest prime number): 36÷2=18
  • Divide 18 by 2: 18÷2=9
  • Divide 9 by 3 (next smallest prime number): 9÷3=3
  • Divide 3 by 3: 3÷3=1

So the prime factorisation of 36 is 2²×3².

  36
 /  \
2    18
    /  \
   2    9
       /  \
      3    3

From the factor tree, you can see that 36 = 2 × 2 × 3 × 3, 2²×3².

Why Prime Factorisation is Useful

Simplifying Fractions:

  • Helps in finding the greatest common divisor.

Finding LCM and GCD:

  • Useful in problems involving least common multiple (LCM) and greatest common divisor (GCD).

Understanding Number Properties:

  • Helps in learning about number divisibility and properties.

Practice Problems

  1. Find the prime factorisation of 30.
  2. Find the prime factorisation of 56.
  3. Write 45 as a product of prime factors.

let’s practice!