Prime Factorisation

Prime factorisation is a method used to find the highest common factor (H.C.F) of two or more numbers. It involves breaking down the numbers into their prime factors and then identifying the common factors.

Steps to find H.C.F using prime factorisation:

1. Prime factorise each number: Find the prime factors of each number by dividing it successively by prime numbers until you obtain only prime factors. For example, let's find the prime factorisation of two numbers, 24 and 36.

- Prime factorisation of 24:
- Divide 24 by the smallest prime number, 2. 24 ÷ 2 = 12.
- Divide 12 by 2 again: 12 ÷ 2 = 6.
- Divide 6 by 2: 6 ÷ 2 = 3.
- Since 3 is a prime number, the prime factorisation of 24 is 2 × 2 × 2 × 3 = 2^3 × 3.

- Prime factorisation of 36:
- Divide 36 by 2: 36 ÷ 2 = 18.
- Divide 18 by 2 again: 18 ÷ 2 = 9.
- Divide 9 by 3: 9 ÷ 3 = 3.
- Since 3 is a prime number, the prime factorisation of 36 is 2^2 × 3^2.

2. Identify the common factors: Compare the prime factors of both numbers and identify the common factors. In this case, the common factors of 24 and 36 are 2 and 3.

3. Multiply the common factors: Multiply all the common factors together to find the H.C.F. In this case, the H.C.F of 24 and 36 is 2 × 3 = 6.

4. Verify: You can verify the obtained H.C.F by checking whether it divides both numbers exactly. In this case, 6 divides both 24 and 36 without leaving any remainder.

Therefore, the H.C.F of 24 and 36 is 6.

Summary:
To find the H.C.F using prime factorisation, we need to prime factorise the given numbers, identify the common factors, multiply them together, and verify if the obtained H.C.F divides the numbers exactly. This method helps us find the highest common factor efficiently by breaking the numbers down into their prime factors.

A.3
b.2