What is gcf

General
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what is gcf

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What is GCF?

Answer:
The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF), is the largest positive integer that divides each of the given integers without leaving a remainder. Understanding GCF is essential in various fields of mathematics, including simplifying fractions, finding least common multiples, and solving problems involving divisibility.

Here’s a detailed breakdown of how to find the GCF:

  1. Prime Factorization Method:

    • Step 1: Find the prime factorization of each number.
      To find the prime factorization, continue dividing the number by prime numbers (2, 3, 5, etc.) until all factors are prime.

    • Step 2: Identify the common prime factors.
      Select the prime factors that appear in the prime factorizations of all numbers.

    • Step 3: Multiply the common prime factors.
      Multiply these common factors together to get the GCF.

    Example:

    Let’s find the GCF of 48 and 60.

    • Prime Factorization of 48:

      48 = 2^4 \times 3

    • Prime Factorization of 60:

      60 = 2^2 \times 3 \times 5

    • Common Prime Factors:

      2^2 \text{ and } 3

    • Multiply the common prime factors:

      GCF = 2^2 \times 3 = 4 \times 3 = 12

    Therefore, the GCF of 48 and 60 is 12.

  2. Division (Euclidean Algorithm) Method:

    • Step 1: Divide the larger number by the smaller number and find the remainder.
      Set the larger number as a and the smaller number as b.

    • Step 2: Replace a with b and b with the remainder from the previous step.

    • Step 3: Repeat the division until the remainder is 0.
      The divisor at this stage will be the GCF.

    Example:

    Let’s find the GCF of 48 and 60 using the Euclidean algorithm.

    • Divide 60 by 48, remainder is 12:

      60 = 48 \times 1 + 12

    • Divide 48 by 12, remainder is 0:

      48 = 12 \times 4 + 0

    The remainder is 0, so the GCF is the divisor at this step, which is 12.

  3. Listing Common Factors Method:

    • Step 1: List all factors of each number.

    • Step 2: Identify the common factors.

    • Step 3: The greatest of these common factors is the GCF.

    Example:

    Let’s find the GCF of 48 and 60 by listing common factors.

    • Factors of 48:

      1, 2, 3, 4, 6, 8, 12, 16, 24, 48

    • Factors of 60:

      1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

    • Common Factors:

      1, 2, 3, 4, 6, 12

    Therefore, the GCF of 48 and 60 is 12.

Final Answer:
The Greatest Common Factor (GCF) of 48 and 60 is 12. This can be determined using any of the methods outlined above.