(5) \( 72 ; 252 \quad \) (6) \( \quad 270 ; 300 \quad \) (7) \( 135 ; 315 \) (9) \( 91 ; 117 ; 231 \quad \) (10) \( \quad 10 ; 92 ; 115 \) Two very large numbers, number A and number B , are decomposed into prime numbers 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. OS is a comparison between two numbers or two quantities with the same unit.

To find the highest common factor (HCF) and the least common multiple (LCM) of the two numbers A and B, we start by examining their prime factorizations. The HCF is calculated by taking the lowest power of each prime factor present in both A and B. For our numbers: - For 2: min(1000, 2000) = 1000 - For 3: min(100, 50) = 50 - For 5: min(20, 30) = 20 - For 7: min(3, 0) = 0 - For 11: min(0, 1) = 0 Thus, the HCF of A and B is: \[ \text{HCF} = 2^{1000} \times 3^{50} \times 5^{20} \] For the LCM, we take the highest power of each prime factor: - 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 A and B is: \[ \text{LCM} = 2^{2000} \times 3^{100} \times 5^{30} \times 7^{3} \times 11^1 \] Did you know that prime factorization has been used since ancient times? The ancient Greeks, particularly Euclid, studied numbers and their properties, which laid the groundwork for number theory as we know it today. His work, "Elements," introduced methods to find the greatest common divisor, a concept that can still be seen in modern mathematics! For those keen on diving deeper into the world of numbers, consider exploring resources such as "Elementary Number Theory" by David M. Burton or “An Introduction to the Theory of Numbers” by G.H. Hardy and E.M. Wright. These texts offer fascinating insights into prime numbers, factors, and various aspects of number theory that can provide a solid foundation for enthusiasts!