Find the LCM and HCF of 6 and 20 by the prime factorisation method.
Step-by-Step Solution
Key Concept: Prime factorisation of numbers; HCF is the product of common prime factors with the smallest exponents, and LCM is the product of all prime factors taken with the highest exponents.
Given: Two numbers 6 and 20.
Step 1: Write the prime factorisation of each number.
$$6 = 2 \times 3$$
$$20 = 2^{2} \times 5$$
Step 2: Identify the common prime factors.
- Common prime factor: 2 (lowest power = $2^{1}$).
Hence, HCF = $2^{1} = 2$.
Step 3: For LCM, take each prime factor appearing in either number with the highest power.
- From 6: $2^{1},\;3^{1}$
- From 20: $2^{2},\;5^{1}$
Highest powers: $2^{2},\;3^{1},\;5^{1}$.
Thus, LCM = $2^{2} \times 3 \times 5 = 4 \times 3 \times 5 = 60$.
Conclusion: HCF(6,20) = 2 and LCM(6,20) = 60.
Correct Answer: HCF = 2, LCM = 60