(9) very large numbers, number A and number B , are decomposed into prime n Tollows: number \( \mathrm{A}=2^{1000} \times 3^{100} \times 5^{20} \times 7^{3} \) umber \( \mathrm{B}=2^{2000} \times 3^{50} \times 5^{30} \times 11 \) Write down the HCF of these numbers, as a product of prime factors. Write down the LCM of these numbers, as a product of prime factors. ( LCM

To find the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of the two numbers \( A \) and \( B \), we start by analyzing their prime factorizations. For number \( A \): \[ A = 2^{1000} \times 3^{100} \times 5^{20} \times 7^{3} \] For number \( B \): \[ B = 2^{2000} \times 3^{50} \times 5^{30} \times 11^{1} \] **HCF** (Highest Common Factor): To calculate the HCF, we take the lowest power for each common prime factor: - For \( 2 \): min(1000, 2000) = \( 1000 \) - For \( 3 \): min(100, 50) = \( 50 \) - For \( 5 \): min(20, 30) = \( 20 \) - For \( 7 \): (not in \( B \), so we ignore it) - For \( 11 \): (not in \( A \), so we ignore it) Thus, the HCF is: \[ \text{HCF}(A, B) = 2^{1000} \times 3^{50} \times 5^{20} \] **LCM** (Lowest Common Multiple): To calculate the LCM, we take the highest power for each prime factor: - For \( 2 \): max(1000, 2000) = \( 2000 \) - For \( 3 \): max(100, 50) = \( 100 \) - For \( 5 \): max(20, 30) = \( 30 \) - For \( 7 \): (only in \( A \), so we take it) - For \( 11 \): (only in \( B \), so we take it) Thus, the LCM is: \[ \text{LCM}(A, B) = 2^{2000} \times 3^{100} \times 5^{30} \times 7^{3} \times 11^{1} \] So the solutions are: - HCF: \( 2^{1000} \times 3^{50} \times 5^{20} \) - LCM: \( 2^{2000} \times 3^{100} \times 5^{30} \times 7^{3} \times 11^{1} \)