(ii) Find the lowest common multiple (LCM) of 54 and \( 3 \times 5^{n} \) in terms of \( n \), where \( n>0 \) Give your answer as the product of its prime factors.

To find the lowest common multiple (LCM) of 54 and \( 3 \times 5^{n} \) in terms of \( n \), we first need to express both numbers in their prime factorized forms. The prime factorization of 54 is: \[ 54 = 2 \times 3^3 \] The prime factorization of \( 3 \times 5^{n} \) is: \[ 3 \times 5^{n} = 3^1 \times 5^n \] The LCM of two numbers is the product of the highest powers of all prime factors that appear in either number. So, we take the highest power of each prime factor that appears in either number: - For the prime factor 2, the highest power is \( 2^1 \) (from 54). - For the prime factor 3, the highest power is \( 3^3 \) (from 54). - For the prime factor 5, the highest power is \( 5^n \) (from \( 3 \times 5^{n} \)). Therefore, the LCM of 54 and \( 3 \times 5^{n} \) is: \[ \text{LCM} = 2^1 \times 3^3 \times 5^n \] Since we are asked to give the answer as the product of its prime factors, the LCM in terms of \( n \) is: \[ \text{LCM} = 2 \times 3^3 \times 5^n \]