(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 determine the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of the two numbers \( \mathrm{A} \) and \( \mathrm{B} \), we'll analyze their prime factorizations. Given: \[ \mathrm{A} = 2^{1000} \times 3^{100} \times 5^{20} \times 7^{3} \] \[ \mathrm{B} = 2^{2000} \times 3^{50} \times 5^{30} \times 11 \] ### Highest Common Factor (HCF) The HCF is found by taking the lowest power of all common prime factors in both numbers. - **Prime 2:** \( \min(1000, 2000) = 1000 \) - **Prime 3:** \( \min(100, 50) = 50 \) - **Prime 5:** \( \min(20, 30) = 20 \) - **Primes 7 and 11:** Not common to both numbers, so they're excluded. Thus, the HCF is: \[ \text{HCF}(\mathrm{A}, \mathrm{B}) = 2^{1000} \times 3^{50} \times 5^{20} \] ### Least Common Multiple (LCM) The LCM is determined by taking the highest power of all prime factors present in either number. - **Prime 2:** \( \max(1000, 2000) = 2000 \) - **Prime 3:** \( \max(100, 50) = 100 \) - **Prime 5:** \( \max(20, 30) = 30 \) - **Prime 7:** Present only in \( \mathrm{A} \), so its power is \( 3 \). - **Prime 11:** Present only in \( \mathrm{B} \), so its power is \( 1 \). Thus, the LCM is: \[ \text{LCM}(\mathrm{A}, \mathrm{B}) = 2^{2000} \times 3^{100} \times 5^{30} \times 7^{3} \times 11 \] ### Summary \[ \boxed{ \begin{aligned} \text{HCF}(\mathrm{A}, \mathrm{B}) &= 2^{1000} \times 3^{50} \times 5^{20} \\ \text{LCM}(\mathrm{A}, \mathrm{B}) &= 2^{2000} \times 3^{100} \times 5^{30} \times 7^{3} \times 11 \end{aligned} } \]