Next, we find the prime factors of 72. Breaking 72 down, we get 2 × 36, then 2 × 2 × 18, then 2 × 2 × 2 × 9, and finally 2 × 2 × 2 × 3 × 3. This can be written in exponential form as 2³ × 3².

Next, we find the prime factors of 72. Breaking 72 down, we get 2 × 36, then 2 × 2 × 18, then 2 × 2 × 2 × 9, and finally 2 × 2 × 2 × 3 × 3. This can be written in exponential form as 2³ × 3².

Now, we find the prime factors of 120. Breaking 120 down, we get 2 × 60, then 2 × 2 × 30, then 2 × 2 × 2 × 15, and finally 2 × 2 × 2 × 3 × 5. This can be written in exponential form as 2³ × 3^1 × 5^1.

Now, we find the prime factors of 120. Breaking 120 down, we get 2 × 60, then 2 × 2 × 30, then 2 × 2 × 2 × 15, and finally 2 × 2 × 2 × 3 × 5. This can be written in exponential form as 2³ × 3^1 × 5^1.

Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

Answer: The HCF of 6, 72 and 120 is 6, and the LCM is 360.

Step-by-step solution

Step 1: Prime Factorisation of 6

To find the HCF and LCM using the prime factorisation method, we first need to express each number as a product of its prime factors. For the number 6, its prime factors are 2 and 3.

Step 2: Prime Factorisation of 72

Next, we find the prime factors of 72. Breaking 72 down, we get $2 \times 36$ , then $2 \times 2 \times 18$ , then $2 \times 2 \times 2 \times 9$ , and finally $2 \times 2 \times 2 \times 3 \times 3$ . This can be written in exponential form as $2^3 \times 3^2$ .

Step 3: Prime Factorisation of 120

Now, we find the prime factors of 120. Breaking 120 down, we get $2 \times 60$ , then $2 \times 2 \times 30$ , then $2 \times 2 \times 2 \times 15$ , and finally $2 \times 2 \times 2 \times 3 \times 5$ . This can be written in exponential form as $2^3 \times 3^1 \times 5^1$ .

Step 4: Calculate HCF

The HCF (Highest Common Factor) is the product of the smallest power of each common prime factor involved in the numbers. The common prime factors are 2 and 3. The smallest power of 2 is $2^1$ (from 6) and the smallest power of 3 is $3^1$ (from 6, 72, and 120). Therefore, the HCF is $2^1 \times 3^1 = 6$ .

Step 5: Calculate LCM

The LCM (Lowest Common Multiple) is the product of the greatest power of each prime factor involved in the numbers. The prime factors involved are 2, 3, and 5. The greatest power of 2 is $2^3$ (from 72 and 120), the greatest power of 3 is $3^2$ (from 72), and the greatest power of 5 is $5^1$ (from 120). Therefore, the LCM is $2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360$ .

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