Q3 Use prime factors to find (i) the HCF and (ii) the LCM of each of the following pairs of numbers. a) 60 and 75 HCF-is b) 54 and 96 c) 108 and 144 d) 200 and 240 e) 168 and 196 f) 150 and 180
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To find the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) using prime factorization, start by breaking each number down into its prime factors. For example, let’s look at the first pair: **a) 60 and 75** - The prime factors of 60 are: 2 × 2 × 3 × 5 (or \(2^2 × 3^1 × 5^1\)). - The prime factors of 75 are: 3 × 5 × 5 (or \(3^1 × 5^2\)). - **HCF**: The common prime factors are 3 and 5, so HCF = 3 × 5 = 15. - **LCM**: Take the highest power of each prime: 2, 3, \(5^2\). So, LCM = 2^2 × 3^1 × 5^2 = 300. Repeat this process for the other pairs: **b) 54 and 96** - 54: \(2^1 × 3^3\) - 96: \(2^5 × 3^1\) - HCF = \(2^1 × 3^1 = 6\) - LCM = \(2^5 × 3^3 = 486\) **c) 108 and 144** - 108: \(2^2 × 3^3\) - 144: \(2^4 × 3^2\) - HCF = \(2^2 × 3^2 = 36\) - LCM = \(2^4 × 3^3 = 432\) **d) 200 and 240** - 200: \(2^3 × 5^2\) - 240: \(2^4 × 3^1 × 5^1\) - HCF = \(2^3 × 5^1 = 40\) - LCM = \(2^4 × 3^1 × 5^2 = 1200\) **e) 168 and 196** - 168: \(2^3 × 3^1 × 7^1\) - 196: \(2^2 × 7^2\) - HCF = \(2^2 × 7^1 = 28\) - LCM = \(2^3 × 3^1 × 7^2 = 1176\) **f) 150 and 180** - 150: \(2^1 × 3^1 × 5^2\) - 180: \(2^2 × 3^2 × 5^1\) - HCF = \(2^1 × 3^1 × 5^1 = 30\) - LCM = \(2^2 × 3^2 × 5^2 = 900\) Now you’ve got the HCF and LCM for each pair of numbers! Prime factorization is like a treasure hunt for numbers, revealing their hidden relationships! Happy calculating!