HCF of \( 5 \times 7 \times 9 \times 11 \times 13 \) and \( 9 \times 11 \times 13 \times 17 \) is equal to \( \begin{array}{ll}\text { (A) } 13 & \text { (B) } 33 \\ \text { (C) } 9 \times 11 \times 13 & \text { (D) } 5 \times 7 \times 9 \times 11 \times 13 \times 17\end{array} \)

To find the highest common factor (HCF) of the two numbers \( 5 \times 7 \times 9 \times 11 \times 13 \) and \( 9 \times 11 \times 13 \), we first need to identify the prime factorization of each number. 1. **Factorization of the first number:** \[ 5 \times 7 \times 9 \times 11 \times 13 \] Here, \( 9 \) can be factored as \( 3^2 \). Thus, the complete factorization is: \[ 5^1 \times 7^1 \times 3^2 \times 11^1 \times 13^1 \] 2. **Factorization of the second number:** \[ 9 \times 11 \times 13 \times 17 \] Again, \( 9 \) can be factored as \( 3^2 \). Therefore, the complete factorization is: \[ 3^2 \times 11^1 \times 13^1 \times 17^1 \] 3. **Finding the HCF:** The HCF is found by taking the lowest power of each prime factor that appears in both factorizations. - For \( 3 \): The minimum power is \( 3^2 \) (appears in both). - For \( 5 \): The minimum power is \( 5^0 \) (does not appear in the second number). - For \( 7 \): The minimum power is \( 7^0 \) (does not appear in the second number). - For \( 11 \): The minimum power is \( 11^1 \) (appears in both). - For \( 13 \): The minimum power is \( 13^1 \) (appears in both). - For \( 17 \): The minimum power is \( 17^0 \) (does not appear in the first number). Therefore, the HCF is: \[ 3^2 \times 11^1 \times 13^1 \] 4. **Calculating the HCF:** \[ HCF = 9 \times 11 \times 13 \] 5. **Conclusion:** The HCF of \( 5 \times 7 \times 9 \times 11 \times 13 \) and \( 9 \times 11 \times 13 \) is \( 9 \times 11 \times 13 \). Thus, the answer is \( \boxed{9 \times 11 \times 13} \).