Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

Step 1: Prime Factorisation of 96

To find the HCF and LCM using the prime factorisation method, we first express each number as a product of its prime factors. For 96, we divide by the smallest prime factor, 2, repeatedly until the remainder is an odd number. Then we divide by the next smallest prime number, 3, until the quotient is 1. This gives us $96 = 2^5 \times 3^1$.

Step 2: Prime Factorisation of 404

Similarly, we find the prime factors of 404. We divide 404 by 2 twice. The remaining factor is 101, which is a prime number. So, $404 = 2^2 \times 101^1$.

Step 3: Calculate HCF

The HCF (Highest Common Factor) is the product of the smallest powers of each common prime factor involved in the numbers. Comparing the prime factorisations of 96 and 404, the only common prime factor is 2. The smallest power of 2 is $2^2$. Therefore, the HCF of 96 and 404 is $2^2 = 4$.

Step 4: Calculate LCM using HCF property

We know that for any two positive integers $a$ and $b$, the product of their HCF and LCM is equal to the product of the numbers themselves (i.e., $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$). We can use this property to find the LCM. Substituting the values, we get $\text{LCM}(96, 404) = \frac{96 \times 404}{4} = 9696$.

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