(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.

Let's analyze the given numbers \( \mathrm{A} \) and \( \mathrm{B} \) to determine their Highest Common Factor (HCF) and Least Common Multiple (LCM). \[ \begin{align*} \mathrm{A} &= 2^{1000} \times 3^{100} \times 5^{20} \times 7^{3} \\ \mathrm{B} &= 2^{2000} \times 3^{50} \times 5^{30} \times 11 \end{align*} \] ### (1) Highest Common Factor (HCF) The HCF is found by taking the minimum exponent for each common prime factor present in both numbers. **Common Prime Factors:** 2, 3, 5 \[ \begin{align*} \text{HCF}(\mathrm{A}, \mathrm{B}) &= 2^{\min(1000, 2000)} \times 3^{\min(100, 50)} \times 5^{\min(20, 30)} \\ &= 2^{1000} \times 3^{50} \times 5^{20} \end{align*} \] ### (2) Least Common Multiple (LCM) The LCM is determined by taking the maximum exponent for each prime factor present in either number. **All Prime Factors:** 2, 3, 5, 7, 11 \[ \begin{align*} \text{LCM}(\mathrm{A}, \mathrm{B}) &= 2^{\max(1000, 2000)} \times 3^{\max(100, 50)} \times 5^{\max(20, 30)} \times 7^{\max(3, 0)} \times 11^{\max(0, 1)} \\ &= 2^{2000} \times 3^{100} \times 5^{30} \times 7^{3} \times 11 \end{align*} \] **Final Answers:** 1. **HCF**: \( 2^{1000} \times 3^{50} \times 5^{20} \) 2. **LCM**: \( 2^{2000} \times 3^{100} \times 5^{30} \times 7^{3} \times 11 \) Answer: 1. HCF = 2¹⁰⁰⁰ × 3⁵⁰ × 5²⁰ 2. LCM = 2²⁰⁰⁰ × 3¹⁰⁰ × 5³⁰ × 7³ × 11