What are the greatest common factors of 8 and 12

General
1

what are the greatest common factors of 8 and 12

2

What are the greatest common factors of 8 and 12?

Answer: The greatest common factor (GCF) of two or more numbers is the largest number that divides all of them without leaving a remainder. To find the GCF of 8 and 12, we can use two methods: the prime factorization method and the Euclidean algorithm. Let’s explore both approaches for clarity.

1. Prime Factorization Method

  1. Prime Factorization of 8:

    8 = 2^3

  2. Prime Factorization of 12:

    12 = 2^2 \times 3

  3. Identify the Common Prime Factors: The common prime factor of 8 and 12 is (2).

  4. Multiply Common Prime Factors with the Lowest Exponents:

    • The lowest exponent of (2) in both prime factorizations is (2).
    \text{GCF} = 2^2 = 4

Thus, the greatest common factor of 8 and 12 is 4.

2. Euclidean Algorithm

  1. Use the Euclidean Algorithm to find the GCF by repeatedly applying the formula:

    \text{GCF}(a, b) = \text{GCF}(b, a \mod b)

    where (a) and (b) are the given numbers.

  2. Step-by-Step Approach:

    • Start with (a = 12) and (b = 8).
    • Calculate (12 \mod 8):
      12 \mod 8 = 4
    • Now, apply the Euclidean Algorithm with (b = 8) and the result (4):
      \text{GCF}(8, 4) = 8 \mod 4 = 0
    • Since the remainder is 0, the GCF is the non-zero divisor, which is 4.

Thus, using both methods, the greatest common factor of 8 and 12 is confirmed to be 4.