Question

Find the HCF and LCM of 96 and 404 by the prime factorisation method and verify that \( \text{HCF} \times \text{LCM} = \) product of the two numbers.

Hint:

For two numbers: - HCF uses minimum powers of common primes. - LCM uses maximum powers of all primes. Always verify using: \( \text{HCF} \times \text{LCM} = a \times b \).

Answer 1

Concept:

• Prime factorisation expresses numbers as a product of prime numbers.
• HCF (Highest Common Factor) = product of common primes with the smallest powers.
• LCM (Least Common Multiple) = product of all primes with the greatest powers.
• Verification property: \[ \text{HCF} \times \text{LCM} = \text{Product of the two numbers} \]

Step 1: Prime factorisation of 96. \[ 96 = 2 \times 48 = 2^2 \times 24 = 2^3 \times 12 = 2^4 \times 6 = 2^5 \times 3 \] \[ \therefore 96 = 2^5 \times 3 \]
Step 2: Prime factorisation of 404. \[ 404 = 2 \times 202 = 2^2 \times 101 \] \[ \therefore 404 = 2^2 \times 101 \]
Step 3: Find the HCF. Common prime factor = \( 2 \) Smallest power of \( 2 \) in both numbers = \( 2^2 \) \[ \text{HCF} = 2^2 = 4 \]
Step 4: Find the LCM. Take highest powers of all primes: \[ \text{LCM} = 2^5 \times 3 \times 101 \] \[ = 32 \times 3 \times 101 \] \[ = 96 \times 101 = 9696 \]
Step 5: Verification. \[ \text{HCF} \times \text{LCM} = 4 \times 9696 = 38784 \] Product of the numbers: \[ 96 \times 404 = 38784 \] \[ \therefore \text{HCF} \times \text{LCM} = \text{Product of the numbers (Verified). \]