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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

Correct Answer: 180

Explanation:

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.


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