Prime factorization

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

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

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How to Find Prime Factorisation

1. Start with the Number:
   • Choose a number you want to factorise.
2. 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.
3. Continue Until the Quotient is a Prime Number:
   • Keep dividing by the smallest prime number until you are left with 1.
4. Write Down the Prime Factors:
   • List all the prime numbers you used to divide the original number.
5. Express as a Product:
   • Write the number as the product of its prime factors.

Example:

Find the Prime Factorisation of 36

1. Start with 36.
2. Divide by 2 (the smallest prime number): 36÷2=18
3. Divide 18 by 2: 18÷2=9
4. Divide 9 by 3 (next smallest prime number): 9÷3=3
5. 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.

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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.

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Why Prime Factorisation is Useful

1. Simplifying Fractions:
   • Helps in finding the greatest common divisor.
2. Finding LCM and GCD:
   • Useful in problems involving least common multiple (LCM) and greatest common divisor (GCD).
3. Understanding Number Properties:
   • Helps in learning about number divisibility and properties.

Learn with an example

let’s practice!