\( A=2^{3} \times 3^{2} \times 5^{4} \) \( B=3^{5} \times 5 \times 7^{3} \) Find the Highest Common Factor (HCF) of \( A \) and \( B \).

To find the Highest Common Factor (HCF) of \( A \) and \( B \), we start by identifying the prime factorization of both expressions. For \( A = 2^{3} \times 3^{2} \times 5^{4} \), the prime factors are: - \( 2 \) with exponent \( 3 \) - \( 3 \) with exponent \( 2 \) - \( 5 \) with exponent \( 4 \) For \( B = 3^{5} \times 5 \times 7^{3} \), the prime factors are: - \( 3 \) with exponent \( 5 \) - \( 5 \) with exponent \( 1 \) - \( 7 \) with exponent \( 3 \) Now we find the HCF by taking the lowest exponent of each common prime factor from both \( A \) and \( B \). - For the prime factor \( 3 \), the minimum exponent is \( \min(2, 5) = 2 \). - For the prime factor \( 5 \), the minimum exponent is \( \min(4, 1) = 1 \). - The prime factor \( 2 \) in \( A \) does not appear in \( B \), and \( 7 \) in \( B \) does not appear in \( A \), so we don't include them. Thus, the HCF of \( A \) and \( B \) is: \[ \text{HCF} = 3^{2} \times 5^{1} = 9 \times 5 = 45 \] The Highest Common Factor (HCF) of \( A \) and \( B \) is \( 45 \).