(f) Two very large numbers, number A and number B , are decomposed into prime numbers as follows: number \( \mathrm{A}=2^{1000} \times 3^{100} \times 5^{20} \times 7^{3} \) number \( \mathrm{B}=2^{2000} \times 3^{50} \times 5^{30} \times 11 \) (1) Write down the HCF of these numbers, as a product of prime factors. (2) Write down the LCM of these numbers, as a product of prime factors.

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the two numbers \( A \) and \( B \), we begin by examining their prime factorizations: Number \( A = 2^{1000} \times 3^{100} \times 5^{20} \times 7^{3} \) Number \( B = 2^{2000} \times 3^{50} \times 5^{30} \times 11^{1} \) (1) To find the HCF, we take the lowest power of each prime factor present in both numbers: - For \( 2 \): min(1000, 2000) = 1000 - For \( 3 \): min(100, 50) = 50 - For \( 5 \): min(20, 30) = 20 - For \( 7 \): not present in \( B = 0\) - For \( 11 \): not present in \( A = 0\) Thus, the HCF of numbers \( A \) and \( B \) is: \[ \text{HCF} = 2^{1000} \times 3^{50} \times 5^{20} \] (2) To find the LCM, we take the highest power of each prime factor present in either number: - For \( 2 \): max(1000, 2000) = 2000 - For \( 3 \): max(100, 50) = 100 - For \( 5 \): max(20, 30) = 30 - For \( 7 \): max(3, 0) = 3 - For \( 11 \): max(0, 1) = 1 Thus, the LCM of numbers \( A \) and \( B \) is: \[ \text{LCM} = 2^{2000} \times 3^{100} \times 5^{30} \times 7^{3} \times 11^{1} \]