\( \begin{array}{l}7 \text { a prime number? Explain. } \\ \text { HCF and LCM for these sets of numbers using prime factors } \\ \text { nd } 56\end{array} \begin{array}{lll}\text { b) } 60 \text { and } 150 & \text { c) } 36 \text { and } 84 .\end{array} \)

Let's break down the problem step by step. ### Part 1: Is 7 a prime number? A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. - The number 7 is greater than 1. - The only divisors of 7 are 1 and 7. Since 7 has exactly two distinct positive divisors, it is indeed a prime number. ### Part 2: Finding HCF and LCM using prime factors To find the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of the given sets of numbers, we will first find their prime factorization. #### a) HCF and LCM of 60 and 150 1. **Prime Factorization**: - \( 60 = 2^2 \times 3^1 \times 5^1 \) - \( 150 = 2^1 \times 3^1 \times 5^2 \) 2. **HCF**: - Take the lowest power of each common prime factor: - For \(2\): \( \min(2, 1) = 1 \) → \(2^1\) - For \(3\): \( \min(1, 1) = 1 \) → \(3^1\) - For \(5\): \( \min(1, 2) = 1 \) → \(5^1\) - Therefore, \( \text{HCF} = 2^1 \times 3^1 \times 5^1 = 30 \) 3. **LCM**: - Take the highest power of each prime factor: - For \(2\): \( \max(2, 1) = 2 \) → \(2^2\) - For \(3\): \( \max(1, 1) = 1 \) → \(3^1\) - For \(5\): \( \max(1, 2) = 2 \) → \(5^2\) - Therefore, \( \text{LCM} = 2^2 \times 3^1 \times 5^2 = 300 \) #### b) HCF and LCM of 36 and 84 1. **Prime Factorization**: - \( 36 = 2^2 \times 3^2 \) - \( 84 = 2^2 \times 3^1 \times 7^1 \) 2. **HCF**: - Take the lowest power of each common prime factor: - For \(2\): \( \min(2, 2) = 2 \) → \(2^2\) - For \(3\): \( \min(2, 1) = 1 \) → \(3^1\) - \(7\) is not common. - Therefore, \( \text{HCF} = 2^2 \times 3^1 = 12 \) 3. **LCM**: - Take the highest power of each prime factor: - For \(2\): \( \max(2, 2) = 2 \) → \(2^2\) - For \(3\): \( \max(2, 1) = 2 \) → \(3^2\) - For \(7\): \( \max(0, 1) = 1 \) → \(7^1\) - Therefore, \( \text{LCM} = 2^2 \times 3^2 \times 7^1 = 252 \) ### Summary of Results - **For 60 and 150**: - HCF = 30 - LCM = 300 - **For 36 and 84**: - HCF = 12 - LCM = 252