Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
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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 2×36, 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.
Step 3: Prime Factorisation of 120
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 23×31×51.
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 21 (from 6) and the smallest power of 3 is 31 (from 6, 72, and 120). Therefore, the HCF is 21×31=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 23 (from 72 and 120), the greatest power of 3 is 32 (from 72), and the greatest power of 5 is 51 (from 120). Therefore, the LCM is 23×32×51=8×9×5=360.