Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
Step-by-Step Solution
Key Concept: Use the prime factorisation method: HCF is the product of the lowest powers of common prime factors; LCM is the product of the highest powers of all prime factors appearing in the numbers.
Given: Numbers are 6, 72 and 120.
To Find: HCF and LCM using prime factorisation.
Step 1 – Prime factorise each number
$$\begin{aligned}
6 &= 2 \times 3 \\
72 &= 2^3 \times 3^2 \\
120 &= 2^3 \times 3 \times 5
\end{aligned}$$
Step 2 – Identify the common prime factors
The primes common to all three numbers are 2 and 3.
Step 3 – Determine the lowest powers of the common primes
- For 2: lowest exponent = $\min\{1,3,3\}=1$
- For 3: lowest exponent = $\min\{1,2,1\}=1$
Step 4 – Compute the HCF
$$\text{HCF}=2^{1}\times 3^{1}=2\times 3=6$$
Step 5 – Determine the highest powers of all primes appearing
- For 2: highest exponent = $\max\{1,3,3\}=3$
- For 3: highest exponent = $\max\{1,2,1\}=2$
- For 5: highest exponent = $\max\{0,0,1\}=1$ (appears only in 120)
Step 6 – Compute the LCM
$$\text{LCM}=2^{3}\times 3^{2}\times 5^{1}=8\times 9\times 5=360$$
Conclusion: HCF = 6 and LCM = 360.
Correct Answer: HCF = 6, LCM = 360