H.C.F of 4 x 27 x 3125, 8 x 9 x 25 x 7 and 16 x 81 x 5 x 11 x 49 is

To find the HCF (Highest Common Factor) of the given numbers, we begin by breaking down each number into its prime factorization. Here are the steps:

1. Prime factorization of 4 × 27 × 3125:
   • 4 = 2²
   
   
   • 27 = 3³
   
   
   • 3125 = 5⁵
   
   So, the prime factorization of 4 × 27 × 3125 is 2² × 3³ × 5⁵.


2. Prime factorization of 8 × 9 × 25 × 7:
   • 8 = 2³
   
   
   • 9 = 3²
   
   
   • 25 = 5²
   
   
   • 7 = 7¹
   
   So, the prime factorization of 8 × 9 × 25 × 7 is 2³ × 3² × 5² × 7¹.


3. Prime factorization of 16 × 81 × 5 × 11 × 49:
   • 16 = 2⁴
   
   
   • 81 = 3⁴
   
   
   • 5 = 5¹
   
   
   • 11 = 11¹
   
   
   • 49 = 7²
   
   So, the prime factorization of 16 × 81 × 5 × 11 × 49 is 2⁴ × 3⁴ × 5¹ × 7² × 11¹.


4. Find the common prime factors and their lowest powers:
   • For 2, the lowest power is 2².
   
   
   • For 3, the lowest power is 3².
   
   
   • For 5, the lowest power is 5¹.
   
   
   • For 7, the lowest power is 7⁰ (since 7 is absent in the first number).
   
   
   • For 11, the lowest power is 11⁰ (since 11 is absent in the first two numbers).


5. Calculate the HCF:
   • HCF = 2² × 3² × 5¹ = 4 × 9 × 5 = 180.


6. Final answer:

   The HCF of the given numbers is 180.